From 9a0246c6d5cb063b99a2677644c9e51d31828065 Mon Sep 17 00:00:00 2001
From: Ken Monks <ken.monks@gmail.com>
Date: Mon, 15 Jul 2024 08:54:25 -0600
Subject: [PATCH] Cleaned Functions library

---
 math/Functions-Rules.lurch | 661 ++++++++++++++++++++++++-------------
 1 file changed, 438 insertions(+), 223 deletions(-)

diff --git a/math/Functions-Rules.lurch b/math/Functions-Rules.lurch
index d5045534..124bb874 100644
--- a/math/Functions-Rules.lurch
+++ b/math/Functions-Rules.lurch
@@ -4,7 +4,7 @@
                 <script language="javascript">
                     const link = document.querySelector( '#loadlink > p > a' )
                     const thisURL = encodeURIComponent( window.location.href )
-                    link?.setAttribute( 'href', 'http://localhost:3000/?load=' + thisURL )
+                    link?.setAttribute( 'href', 'http://localhost:2999/instructor.html?load=' + thisURL )
                 </script>
             </div>
         
@@ -14,181 +14,250 @@
                 
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+        
+            <div id="metadata" style="display: none;"></div>
+            <div id="document"><h1>Math 299 Rules, Definitions, and Theorems</h1>
+<h3>Ken Monks - University of Scranton -&nbsp;Spring 2024</h3>
+<p>Welcome to Math 299.&nbsp; Below you will find some of the rules, definitions, theorems, and declarations that we use in the course.&nbsp; For more detail about these rules see the Course Lecture Notes on the Math 299 Home Page accessible via the Lurch <code>Help</code> menu at the top of the page.&nbsp; (<em>Note that some of the definitions below differ from what is in the lecture notes as they are still in the process of being updated to be consistent with the rules below</em>.)</p>
+<p>There are many ways to write the same definition, theorem, or rule in Lurch.&nbsp; For consistency most of the rules below use the same style that imitates the <em>Given-Conclude</em> format used in many high school geometry textbooks.</p>
+<p>&nbsp;</p></div>
+        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Filename:</td><td><tt>math/Intro.lurch</tt></td></tr><tr><td>Source:</td><td>the web</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div><div data-category="settings" data-key="shell style" data-value-type="json">"boxed"</div></div>
             <div id="document"><h1>Propositional Logic</h1>
-<p>This is a basic library of the rules that define a natural deduction style formal system for Propositional Logic. Premises are denoted by 'From' and conclusions by 'Conclude'.</p>
+<p>This is a basic library of the rules that define a natural deduction style formal system for Propositional Logic. Rule premises are denoted as 'Given' and conclusions by 'Conclude'.</p>
 <hr>
 <h3><strong>Constants</strong></h3>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare ⇔, and, ⇒, or, ¬, contradiction&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">⇔</span><span class="ML__text">, and, </span><span class="ML__cmr">⇒</span><span class="ML__text">, or, </span><span class="ML__ams">¬</span><span class="ML__text">, and </span><span class="ML__cmr">→←</span></span></span></span></span> to be constants.</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare iff, and, implies, or, not, contradiction&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">⇔</span><span class="ML__text">, and, </span><span class="ML__cmr">⇒</span><span class="ML__text">, or, </span><span class="ML__ams">¬</span><span class="ML__text">, and </span><span class="ML__cmr">→←</span></span></span></span></span> to be constants.</p>
 <h3><strong>Definitions</strong></h3>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p style="text-align: left;"><strong>and+</strong></p>
-<p style="text-align: left;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W and V&quot;" data-metadata_latex="&quot; W\\text{ and } V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></em></p>
+</div>
+<p style="text-align: left;"><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W and V&quot;" data-metadata_latex="&quot; W\\text{ and } V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>and-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W and V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;" data-metadata_latex="&quot;W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W and V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+</div>
+<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;" data-metadata_latex="&quot;W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>implies+</strong></p>
-<p>From the subproof</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+</div>
 </div>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V&quot;" data-metadata_latex="&quot; W\\Rightarrow V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V&quot;" data-metadata_latex="&quot; W\\Rightarrow V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>implies- (modus ponens)</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W implies V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+</div>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>iff+</strong></p>
-<p>From the two subproofs:</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span>&nbsp;</p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
-<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;" data-metadata_latex="&quot;W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W iff V&quot;" data-metadata_latex="&quot; W\\Leftrightarrow V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+</div>
+<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W iff V&quot;" data-metadata_latex="&quot; W\\Leftrightarrow V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>iff- (1 of 2)</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W iff V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W iff V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+</div>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;" data-metadata_latex="&quot;V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>iff- (2 of 2)</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W iff V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W iff V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+</div>
 <p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>or+</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V&quot;" data-metadata_latex="&quot; W\\text{ or } V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V or W&quot;" data-metadata_latex="&quot; V\\text{ or } W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+</div>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V&quot;" data-metadata_latex="&quot; W\\text{ or } V&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V or W&quot;" data-metadata_latex="&quot; V\\text{ or } W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>or- (proof by cases)</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W implies U&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From V implies U&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;U&quot;" data-metadata_latex="&quot;U&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span>&nbsp;</p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies U&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V implies U&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span></p>
+</div>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;U&quot;" data-metadata_latex="&quot;U&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span>&nbsp;</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>not+ (proof by contradiction)</strong></p>
-<p>From the subproof:</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;contradiction&quot;" data-metadata_latex="&quot;\\rightarrow\\leftarrow&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;contradiction&quot;" data-metadata_latex="&quot;\\rightarrow\\leftarrow&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
+</div>
 </div>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W&quot;" data-metadata_latex="&quot;\\neg W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W&quot;" data-metadata_latex="&quot;\\neg W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>not- (proof by contradiction)</strong></p>
-<p>From the subproof:</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;contradiction&quot;" data-metadata_latex="&quot;\\rightarrow\\leftarrow&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Assume not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;contradiction&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
 </div>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;" data-metadata_latex="&quot;W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+</div>
+<p>&nbsp;Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;" data-metadata_latex="&quot;W&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>contradiction+</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;→←&quot;" data-metadata_latex="&quot;\\rightarrow\\leftarrow&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+</div>
+<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;→←&quot;" data-metadata_latex="&quot;\\rightarrow\\leftarrow&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
 </div>
 <hr>
 <p>&nbsp;</p></div>
-        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Prop-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
+        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Filename:</td><td><tt>math/Prop-Rules.lurch</tt></td></tr><tr><td>Source:</td><td>the web</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
             <div id="document"><h1>Predicate Logic with Equality</h1>
 <p>This is a basic library of the rules that define a natural deduction style formal system for Predicate Logic with equality.&nbsp; All of the rules from the Propositional Logic library are also available.</p>
 <hr>
 <div>
 <h3>Constants</h3>
-<div><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare ∀, ∃, ∃!, =&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">∀</span><span class="ML__text">, </span><span class="ML__cmr">∃</span><span class="ML__text">, </span><span class="ML__cmr">∃!</span><span class="ML__text">, and </span><span class="ML__cmr">=</span></span></span></span></span> to be constants.</div>
+<div><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare forall, exists, exists!, =&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">∀</span><span class="ML__text">, </span><span class="ML__cmr">∃</span><span class="ML__text">, </span><span class="ML__cmr">∃!</span><span class="ML__text">, and </span><span class="ML__cmr">=</span></span></span></span></span> to be constants.</div>
 <h3>Quantifier Rules</h3>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong style="font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Open Sans', 'Helvetica Neue', sans-serif;">forall+</strong></p>
-<p>From the subproof:</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(a variable declaration)&nbsp;</p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(a variable declaration)</p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(z)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;forall x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∀</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;forall x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∀</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>forall-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From forall x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cmr">∀</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;forall x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∀</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+</div>
+<p><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>exists+</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From 𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
-<div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>exists-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From exists x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(c) for some c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">c</span></span></span></span></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (constant declaration)</p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></em></p>
 </div>
-<h3>Equality</h3>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(c) for some c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">c</span></span></span></span></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (constant declaration)</p>
 </div>
+<h3>Equality</h3>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
@@ -196,35 +265,45 @@
 </div>
 </div>
 <div>
-<p><strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;SubsRule&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Substitution </span></span></span></span></span>Rule:</strong></p>
+<p><strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;SubsRule&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Substitution </span></span></span></span></span>Rule&nbsp;</strong></p>
+<p>Note: For efficiency, most documents will not use this rule unless you explicitly say 'by substitution' in a meaningful expression after the statement you are trying to justify.</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From w=v&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.03em;">w</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">v</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From 𝜆P(w)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.03em;">w</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(v)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">v</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
+<p><strong>Substitution</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;w=v&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.03em;">w</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">v</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(w)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.03em;">w</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(v)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">v</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
 </div>
 </div>
 <h3>Unique Existence</h3>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>unique existence-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From ∃!x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cmr">∃!</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(c) and (forall y.(𝜆P(y) implies y=c)) for some c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">∀</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">.</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">c</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom atom-is-selected" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists!x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃!</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(c) and (forall y.(𝜆P(y) implies y=c)) for some c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">∀</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">.</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">c</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>unique existence+</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From 𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
-<p>From the subproof:&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(t)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.61508em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">t</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let y be such that 𝜆P(y)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> be such that </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;y=t&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit">t</span></span></span></span></span></p>
 </div>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;∃!x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃!</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists!x.𝜆P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃!</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
 </div>
 <hr></div>
 </div></div>
-        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Pred-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
+        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Filename:</td><td><tt>math/Pred-Rules.lurch</tt></td></tr><tr><td>Source:</td><td>the web</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
             <div id="document"><div>
 <div>&nbsp;</div>
 </div>
@@ -232,145 +311,166 @@
 <p>This is a basic library of the rules that are theorems provable in the system defined by the rules in the Predicate Logic with Equality library.</p>
 <hr>
 <h3>Constants</h3>
-<p>No additional constants are required.</p>
+<p>No additional constants are required.&nbsp;</p>
 <h3>Theorems (as rules)</h3>
-<div>
-<div>
+<p>The first rule uses '<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span>' to indicate that it functions effectively as two rules, one where&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\text{not}(\\text{not}(W))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not</span><span class="ML__cmr">(</span><span class="ML__text">not</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">))</span></span></span></span></span> is given and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span> is the conclusion and a second rule where <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span> is given and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\text{not}(\\text{not}(W))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not</span><span class="ML__cmr">(</span><span class="ML__text">not</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">))</span></span></span></span></span> is the conclusion.&nbsp; The analogous interpretation applies even when there are multiple expressions on the left hand or right hand side of the <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> symbol.</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>excluded middle</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><strong>Theorem (double negative)</strong></p>
+<p style="text-align: left;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (not W) equiv W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span>&nbsp; &nbsp;</p>
 </div>
+<div>
+<div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>double negative</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (not W)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: -0.2em; height: 0.8999999999999999em;"><span class="ML__small-delim ML__open">(</span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><strong>Theorem (excluded middle)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>commutativity of 'or'</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W or V &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
-<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V or W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><strong>Theorem (commutativity of 'or')</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+</div>
+<p><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V or W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>associativity of 'or'</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(W or V) or U ≡ W or (V or U)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__cmr">≡</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Theorem (associativity of 'or')</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(W or V) or U ≡ W or (V or U)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
 </div>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>alternate definition of 'implies'</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (alternate definition of 'implies')</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V equiv &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>alternate or-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (alternate or-)</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></em></p>
+</div>
+<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>alternate or-</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (alternate or-)</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W or V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></em></p>
+</div>
 <p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>negated implication</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W implies V)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp; &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;≡&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span>&nbsp; &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span> and&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (negated implication)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W implies V)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;≡&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span>,&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>contrapositive</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span>&nbsp; &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;≡ &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span>&nbsp; &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not V implies not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">⇒</span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><strong>Theorem (contrapositive)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W implies V equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot; not V implies not W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">⇒</span><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>DeMorgan's Law</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W and V)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> &nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> &nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W or not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (DeMorgan's Law)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W and V) equiv &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W or not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or not </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>DeMorgan's Law</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W or V)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> &nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> &nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W and not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__ams">¬</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
+<p><strong>Theorem (DeMorgan's Law)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (W or V) equiv &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not W and not V&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and not </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Anything follows from a contradiction</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From →←&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__cmr">→←</span></span></span></span></span>&nbsp;</p>
-<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
+<p><strong>Theorem (anything follows from a contradiction)</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;→←&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></em></p>
+</div>
+<p>Conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;W&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
 </div>
 <hr></div>
 </div></div>
         </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Logic-theorems-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
-            <div id="document"><div>
-<div>
-<div>
-<h1>Peano Axioms</h1>
+            <div id="document"><h1>Peano Axioms</h1>
 <div>This is a library containing the Peano Axioms. They define a formal system for the Natural Numbers.&nbsp; This library includes all of the rules, definitions, and axioms from the Logic Theorems library.</div>
 <hr>
 <h3>Constants</h3>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare 0, sigma, +, ⋅, ≤&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">0</span><span class="ML__text">, </span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__text">, </span><span class="ML__cmr">+</span><span class="ML__text">, </span><span class="ML__cmr">⋅</span><span class="ML__text">, and </span><span class="ML__cmr">≤</span></span></span></span></span> to be constants.</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare 0, sigma, +, *, leq&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">0</span><span class="ML__text">, </span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__text">, </span><span class="ML__cmr">+</span><span class="ML__text">, </span><span class="ML__cmr">⋅</span><span class="ML__text">, and </span><span class="ML__cmr">≤</span></span></span></span></span> to be constants.</p>
 <h3>Peano Axioms</h3>
-<h4><strong>The set ℕ</strong></h4>
-<p>&nbsp; &nbsp; (We don't need axioms N0 or N1.)</p>
-<p>&nbsp; &nbsp; <strong>N0:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;0&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span></span></span></span></span> is a natural number.</p>
-<p>&nbsp; &nbsp; <strong>N1:</strong> For all natural numbers&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;n&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span></span></span></span></span>, the successor of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;n&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span></span></span></span></span> is a natural number.</p>
+<h4><strong>The set <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\mathbb{N}&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__bb">N</span></span></span></span></span></strong></h4>
+<p>(We don't need axioms N0 or N1 because in this context the domain of discourse is the set of natural numbers.)</p>
+<p>&nbsp; &nbsp; <strong>Axiom N0:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;0&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span></span></span></span></span> is a natural number.</p>
+<p>&nbsp; &nbsp; <strong>Axiom </strong><strong>N1:</strong> For all natural numbers&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;n&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span></span></span></span></span>, the successor of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;n&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span></span></span></span></span> is a natural number.</p>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>N2 (successor is injective):&nbsp; &nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If sigma(m)=sigma(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
+<p><strong>Axiom N2 (successor is injective)</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given </em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sigma(m)=sigma(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>N3 (zero is first):&nbsp; &nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0 neq sigma(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Axiom N3 (zero is first)&nbsp; &nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0 neq sigma(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>N4 (induction):</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If 𝜆P(0)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (<em>the Base Case</em>)</p>
-<p>and we have the subproof&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <em>(the Inductive step)</em></p>
+<p><strong>Axiom N4 (induction)</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let k be such that 𝜆P(k)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__text"> be such that </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(0)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> (base case)</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let k be such that 𝜆P(k)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__text"> be such that </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> (<em>inductive hypothesis</em>)</p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;𝜆P(sigma(k))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p>then we can conclude&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;forall n.𝜆P(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∀</span><span class="ML__mathit">n</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+</div>
+<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;forall n.𝜆P(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∀</span><span class="ML__mathit">n</span><span class="ML__cmr">.</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div><strong>Addition</strong></div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>A0:&nbsp; &nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n+0=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__cmr">0</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
+<p><strong>Axiom A0 (base case for addition)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n+0=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__cmr">0</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>A1:&nbsp; &nbsp;&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m+sigma(n)=sigma(m+n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: -0.08333em; height: 0.66666em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Axiom A1 (recursion for addition)&nbsp; &nbsp;&nbsp;</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m+sigma(n)=sigma(m+n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: -0.08333em; height: 0.66666em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div><strong>Multiplication</strong></div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>M0: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n*0=0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__cmr">0</span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
+<p><strong>Axiom M0 (base case for multiplication)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n*0=0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__cmr">0</span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>M1: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*sigma(n) = m+m*n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span></span></span></span></span></p>
+<p><strong>Axiom M1 (recursion for multiplication)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*sigma(n) = m+m*n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span></span></span></span></span></p>
 </div>
 </div>
 <div><strong>Order</strong></div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>I: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m leq n ≡ exists k.m+k=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">≤</span><span class="ML__mathit">n</span><span class="ML__cmr">≡</span><span class="ML__cmr">∃</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">.</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
+<p><strong>Axiom I (inequality)</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m leq n equiv exists k.m+k=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">≤</span><span class="ML__mathit">n</span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__cmr">∃</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">.</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
 </div>
 </div>
 <hr></div>
-</div>
-</div></div>
         </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Peano-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
             <div id="document"><div>
 <div>&nbsp;</div>
@@ -379,7 +479,7 @@
 <p>This library contains some definitions from Number Theory.&nbsp; It includes all of the rules in the Peano Axioms library.</p>
 <hr>
 <h3>Constants</h3>
-<p><span class="lurch-atom atom-is-selected" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare 1, 2, 3, 4, 5, <, is, |, prime, even, odd&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">1</span><span class="ML__text">, </span><span class="ML__cmr">2</span><span class="ML__text">, </span><span class="ML__cmr">3</span><span class="ML__text">, </span><span class="ML__cmr">4</span><span class="ML__text">, </span><span class="ML__cmr">5</span><span class="ML__text">, </span><span class="ML__cmr">&lt;</span><span class="ML__text">, is, </span><span class="ML__cmr">∣</span><span class="ML__text">, prime, even, and odd</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare 1, 2, 3, 4, 5, <, is, |, prime, even, odd&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">1</span><span class="ML__text">, </span><span class="ML__cmr">2</span><span class="ML__text">, </span><span class="ML__cmr">3</span><span class="ML__text">, </span><span class="ML__cmr">4</span><span class="ML__text">, </span><span class="ML__cmr">5</span><span class="ML__text">, </span><span class="ML__cmr">&lt;</span><span class="ML__text">, is, </span><span class="ML__cmr">∣</span><span class="ML__text">, prime, even, and odd</span></span></span></span></span></p>
 <h3>Definitions</h3>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Base ten digits</strong><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1=sigma(0)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
@@ -389,20 +489,21 @@
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;5=sigma(4)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">5</span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">4</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Less than:</strong> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m<n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m leq n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">≤</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp; &amp;&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m neq n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
+<p><strong>Less than</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m<n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m leq n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">≤</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp; ,&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m neq n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
 </div>
 <div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Divides:&nbsp; &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If m|n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">m</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n=k*m for some k&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__text"> for some </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span></span></span></span></span>.&nbsp;</p>
+<p><strong>Prime&nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n is prime&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__text"> is prime</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;≡&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span>&nbsp; <span class="lurch-atom atom-is-selected" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1<n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp; ,&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (exists k.(1<k and k<n) and k|n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">∃</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">.</span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.73354em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">1</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
+<p>We can also write rules using the 'If-then' style as an alternative to the 'Given-Conclude' style used in this document's context.&nbsp; We will use this form from now on for any rule that does not contain a subproof premise, or the <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span> rule shorthand.</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Divides:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If n=k*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m|n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span></span></span></span></span>.</p>
-</div>
+<p><strong>Divides:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If m|n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">m</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n=k*m for some k&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__text"> for some </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span></span></span></span></span>.&nbsp;</p>
 </div>
-<div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Prime: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n is prime&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__text"> is prime</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;≡&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1<n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp; &amp;&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (exists k.(1<k and k<n) and k|n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">∃</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">.</span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.73354em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">1</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Divides:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If n=k*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m|n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">∣</span><span class="ML__mathit">n</span></span></span></span></span>.</p>
 </div>
 </div>
 <div>
@@ -415,7 +516,8 @@
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Odd:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n is odd&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__text"> is odd</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">≡</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not n is even&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__mathit">n</span><span class="ML__text"> is even</span></span></span></span></span></p>
+<p><strong>Odd </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n is odd&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__text"> is odd</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not n is even&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit">n</span><span class="ML__text"> is even</span></span></span></span></span></p>
 </div>
 <hr></div>
 </div></div>
@@ -425,9 +527,10 @@
 <p>&nbsp;</p>
 <p style="text-align: center;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\begin{align*} a &amp;= b \\\\ &amp;=c \\\\ &amp;=d&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mtable"><span class="col-align-r"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 2.36em;"><span style="top: -4.51em;"><span style="height: 1.5em; display: inline-block;"><span class="ML__mathit">a</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="col-align-l"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 2.36em;"><span style="top: -4.51em;"><span style="height: 1.5em; display: inline-block;"><span class="ML__cmr">=</span><span class="ML__mathit">b</span></span></span><span style="top: -3.01em;"><span style="height: 1.5em; display: inline-block;"><span class="ML__cmr">=</span><span class="ML__mathit">c</span></span></span><span style="top: -1.51em;"><span style="height: 1.2em; display: inline-block;"><span class="ML__cmr">=</span><span class="ML__mathit">d</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span></span></span></span></span></span></p>
 <p style="text-align: left;">&nbsp;</p>
-<p style="text-align: left;">(entered as&nbsp;<code>a=b=c=d</code>).&nbsp; It automates the application of the substitution and reflexive rules of equality, its derived properties of symmetry and transitivity, and checks each pair in the chain for both direct equality, and equality via a single substitution.&nbsp; This topic includes everything from the Number Theory topic.</p>
+<p style="text-align: left;">(entered as&nbsp;<code>a=b=c=d</code>).&nbsp; It automates the application of the substitution and reflexive rules of equality, the derived properties of symmetry and transitivity, and checks each pair in the chain for both direct equality, and equality via a single substitution.&nbsp; This topic includes everything from the Number Theory topic.</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Equations Rule:</strong> &nbsp; <span class="lurch-atom atom-is-selected" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;EquationsRule&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">EquationsRule</span></span></span></span></span></p>
+<p><strong>Equations Rule</strong>&nbsp; &nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;EquationsRule&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">EquationsRule</span></span></span></span></span></p>
 </div>
 <hr>
 <p>&nbsp;</p></div>
@@ -437,44 +540,55 @@
 <hr>
 <h3>Theorems</h3>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (alternate definition of 𝜎):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sigma(n)=n+1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><strong>Theorem (alternate definition of 𝜎)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sigma(n)=n+1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (commutativity of addition):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m+n=n+m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span></span></span></span></span></p>
+<p><strong>Theorem (commutativity of addition)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m+n=n+m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (associativity of addition):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(m+n)+p = m+(n+p)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.08333em; height: 0.66666em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Theorem (associativity of addition)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(m+n)+p = m+(n+p)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.08333em; height: 0.66666em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (addition of zero on the left):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0+n=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
+<p><strong>Theorem (addition of zero on the left)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0+n=n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">+</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (multiplication by zero on the left):</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0*n = 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
+<p><strong>Theorem (multiplication by zero on the left)</strong>&nbsp;</p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0*n = 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (identity of multiplication):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n*1 = n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__cmr">1</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
+<p><strong>Theorem (identity of multiplication)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n*1 = n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__cmr">1</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (identity of multiplication):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1*n = n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
+<p><strong>Theorem (identity of multiplication)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1*n = n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span></span></span></span></span>&nbsp;</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (commutativity of multiplication):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*n = n*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span></p>
+<p><strong>Theorem (commutativity of multiplication)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*n = n*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem(associativity of multiplication):&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(m*n)*p = m*(n*p)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.44445em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.63889em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Theorem(associativity of multiplication)&nbsp;</strong></p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(m*n)*p = m*(n*p)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.44445em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.63889em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (distributive property of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\cdot&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">⋅</span></span></span></span></span> over <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;+&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">+</span></span></span></span></span>):</strong> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*(n+p)=m*n+m*p&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span></span></span></span></span></p>
+<p><strong>Theorem (distributive property of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\cdot&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">⋅</span></span></span></span></span> over <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;+&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">+</span></span></span></span></span>)</strong>&nbsp;&nbsp;</p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;m*(n+p)=m*n+m*p&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span><span class="ML__cmr">⋅</span><span class="ML__mathit">p</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Theorem (distributive property of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\cdot&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">⋅</span></span></span></span></span> over <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;+&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">+</span></span></span></span></span>):</strong> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(n+p)*m=n*m+p*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span></p>
+<p><strong>Theorem (distributive property of <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;\\cdot&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">⋅</span></span></span></span></span> over <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;+&quot;" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">+</span></span></span></span></span>)</strong>&nbsp;&nbsp;</p>
+<p style="text-align: center; line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(n+p)*m=n*m+p*m&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7777700000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__cmr">=</span><span class="ML__mathit">n</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span><span class="ML__cmr">+</span><span class="ML__mathit">p</span><span class="ML__cmr">⋅</span><span class="ML__mathit">m</span></span></span></span></span></p>
 </div>
 </div>
 <hr>
@@ -484,68 +598,78 @@
 <p>Once we have defined the natural numbers we can use them to define sequences, both finite and infinite. They can be defined explicitly or recursively.&nbsp; In this topic we define a few well known recursive sequences.&nbsp; All of the rules, definitions, and theorems in the Number Theory Theorems topic are also available here.</p>
 <hr>
 <h3>Constants</h3>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare Sum, Fib, !, multinomial, choose, ^&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare Sum, </span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__text">, factorial, multinomial, choose, and </span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.81em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.39em; display: inline-block; font-size: 70%;"><span class="ML__cmr">∧</span></span></span></span></span></span></span></span></span></span></span> to be constants.</p>
+<p><span class="lurch-atom atom-is-selected" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare Sum, Fib, !, multinomial, choose, ^&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare Sum, </span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__text">, factorial, multinomial, choose, and </span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.81em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.39em; display: inline-block; font-size: 70%;"><span class="ML__cmr">∧</span></span></span></span></span></span></span></span></span></span></span> to be constants.</p>
 <h3>Definitions</h3>
-<p><strong>Summation:</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sum k to 0 of @P(k) = @P(0)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.81em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.3em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">0</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Summation (base case)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sum k to 0 of @P(k) = @P(0)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.81em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.3em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">0</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.64444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sum k to sigma(n) of @P(k) = @P(sigma(n))+sum k to n of @P(k)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.97em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.38em;"><span style="height: 0.7em; display: inline-block; font-size: 70%;"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">+</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.66em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.3em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Summation (recursion)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sum k to sigma(n) of @P(k) = @P(sigma(n))+sum k to n of @P(k)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.97em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.38em;"><span style="height: 0.7em; display: inline-block; font-size: 70%;"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">+</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.66em;"><span class="ML__center" style="top: -1.84em;"><span style="height: 0.49em; display: inline-block; font-size: 70%;"><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__cmr">=0</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">∑</span></span></span><span class="ML__center" style="top: -4.3em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">k</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p><strong>Natural Number Exponentiation:</strong></p>
 <div>
+<p>&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z^0=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">0</span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><strong>Natural Number Exponentiation (base case)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z^0=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">0</span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z^(sigma(n))=z*z^n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.94em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.7em; display: inline-block; font-size: 70%;"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.72em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span></span></span></span></span></span></span></span></p>
+<p><strong>Natural Number Exponentiation (recursion)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z^(sigma(n))=z*z^n&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.94em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.7em; display: inline-block; font-size: 70%;"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.72em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span></span></span></span></span></span></span></span></p>
 </div>
-<p><strong>Factorial:</strong></p>
+<p>&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0! =1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0!</span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><strong>Factorial (base case)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0! =1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0!</span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sigma(n)! = sigma(n)*n!&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">!</span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">!</span></span></span></span></span></p>
+<p><strong>Factorial (recursion)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;sigma(n)! = sigma(n)*n!&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">!</span><span class="ML__cmr">=</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">n</span><span class="ML__cmr">!</span></span></span></span></span></p>
 </div>
 </div>
 <div>
-<p><strong>Fibonacci Numbers:</strong></p>
+<p>&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
+<p><strong>Fibonacci Numbers (base cases)</strong></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Fib_(0)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">0</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Fib_(1)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.46em; display: inline-block; font-size: 70%;"><span class="ML__cmr">1</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Fib_(n+2)=Fib_(n+1)+Fib_(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span><span class="ML__cmr">+2</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span><span class="ML__cmr">+1</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.16em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span></span></span></span></span></p>
+<p><strong>Fibonacci Numbers (recursion)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Fib_(n+2)=Fib_(n+1)+Fib_(n)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span><span class="ML__cmr">+2</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.31em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span><span class="ML__cmr">+1</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.14em;">F</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.16em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.31em; display: inline-block; font-size: 70%;"><span class="ML__mathit">n</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span></span></span></span></span></p>
 </div>
 </div>
 <div>
-<p><strong>Multinomial coefficients:</strong></p>
+<p>&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;multinomial(m,0)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8388800000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">,</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><strong>Multinomial coefficients (base case)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;multinomial(m,0)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8388800000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">,</span><span class="ML__cmr">0</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;multinomial(0,n)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8388800000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0,</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><strong>Multinomial coefficients (base case)</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;multinomial(0,n)=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8388800000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">0,</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
+<p><strong>Multinomial coefficients (recursion)</strong></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;multinomial(sigma(m),sigma(n)) \n  = multinomial(sigma(m),n) +\n    multinomial(m,sigma(n))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">,</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">,</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">+</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">m</span><span class="ML__cmr">,</span><span class="lcGreek ML__mathit" style="margin-right: 0.04em;">σ</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div>
-<p><strong>Binomial Coefficients:</strong></p>
+<p>&nbsp;</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n+m choose m =multinomial(n,m)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mfrac"><span class="ML__delim-size3">(</span><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.09em;"><span class="ML__center" style="top: -3.5em;"><span style="height: 0.67em; display: inline-block;"><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span></span></span><span class="ML__center" style="top: -2.31em;"><span style="height: 0.44em; display: inline-block;"><span class="ML__mathit">m</span></span></span></span><span class="ML__vlist-s">​</span></span></span><span class="ML__delim-size3">)</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">,</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Binomial Coefficients</strong></p>
+<p style="line-height: 2;"><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;n+m choose m =multinomial(n,m)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mfrac"><span class="ML__delim-size3">(</span><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.09em;"><span class="ML__center" style="top: -3.5em;"><span style="height: 0.67em; display: inline-block;"><span class="ML__mathit">n</span><span class="ML__cmr">+</span><span class="ML__mathit">m</span></span></span><span class="ML__center" style="top: -2.31em;"><span style="height: 0.44em; display: inline-block;"><span class="ML__mathit">m</span></span></span></span><span class="ML__vlist-s">​</span></span></span><span class="ML__delim-size3">)</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">n</span><span class="ML__cmr">,</span><span class="ML__mathit">m</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
 <div><hr></div></div>
         </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Sequences-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
             <div id="document"><div>
-<div>
 <h1>Real Numbers&nbsp;</h1>
 </div>
 <p>We can now define a formal system for the Real Numbers.&nbsp; This library includes all of the rules, definitions, and axioms from the Sequences library.</p>
 <h3>Constants</h3>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare <, -&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">&lt;</span><span class="ML__text"> and </span><span class="ML__cmr">−</span></span></span></span></span> (the negative sign) to be constants.</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare -&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">−</span></span></span></span></span> (the negative sign) to be constants.</p>
 <h3>Axioms</h3>
 <p>We include here all of the rules of inference for the real numbers except for the completeness axiom, which is beyond the scope of this course.</p>
 <h4><strong>Axioms of Addition</strong></h4>
@@ -557,19 +681,23 @@
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x+(-x)=0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__left-right" style="margin-top: -0.08333em; height: 0.66666em;"><span class="ML__small-delim ML__open">(</span><span class="ML__cmr">−</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;-x+x=0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">−</span><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
 </div>
+<p>The next two axioms (and several more below) are already available as theorems about the natural numbers, so we don't repeat them here.&nbsp; The only difference is now we have changed our domain of discourse from the natural numbers to the real numbers.</p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Commutativity of addition: </strong>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x+y=y+x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit">x</span></span></span></span></span></p>
+<p><strong>Commutativity of addition</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x+y=y+x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit">x</span></span></span></span></span></p>
 </div>
 <div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Associativity of addition:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;(x+y)+z = x+(y+z)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">(</span><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">)</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span></span></span></span></span></p>
+<p><strong>Associativity of addition</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;(x+y)+z = x+(y+z)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">(</span><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">)</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span></span></span></span></span></p>
 </div>
 </div>
 <h4>Axioms of Multiplication</h4>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Multiplicative identity&nbsp;</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1 neq 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
+<p><strong>Multiplicative identity&nbsp;</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;1 neq 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">1</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Multiplicative identity</strong></p>
@@ -578,17 +706,17 @@
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Multiplicative inverse</strong></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;if x neq 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">if </span><span class="ML__mathit">x</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span>&nbsp; then</p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x*x^-=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.83em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__cmr">−</span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span>&nbsp;</p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(x^-)*x=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.821331em;"><span class="ML__open ML__delim-size1">(</span><span class="ML__mathit">x</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.83em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__cmr">−</span></span></span></span></span></span></span><span class="ML__close ML__delim-size1">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;if x neq 0&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">if </span><span class="ML__mathit">x</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">0</span></span></span></span></span>&nbsp; then&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x*x^-=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.83em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__cmr">−</span></span></span></span></span></span></span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(x^-)*x=1&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.821331em;"><span class="ML__open ML__delim-size1">(</span><span class="ML__mathit">x</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.83em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__cmr">−</span></span></span></span></span></span></span><span class="ML__close ML__delim-size1">)</span></span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__cmr">1</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Commutativity of multiplication:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x\\cdot y=y\\cdot x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span></span></span></span></span></p>
+<p><strong>Commutativity of multiplication&nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x\\cdot y=y\\cdot x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">⋅</span><span class="ML__mathit">x</span></span></span></span></span></p>
 </div>
 <div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Associativity of multiplication:&nbsp; &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;(x\\cdot y)\\cdot z = x\\cdot (y\\cdot z)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">(</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">)</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span></span></span></span></span></p>
+<p><strong>Associativity of multiplication</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;(x\\cdot y)\\cdot z = x\\cdot (y\\cdot z)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">(</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">)</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span></span></span></span></span></p>
 </div>
 </div>
 </div>
@@ -596,36 +724,44 @@
 <h4>Axioms of Order</h4>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>&lt; is irreflexive:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (x<x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">¬</span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.5782em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>&lt; is irreflexive&nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;not (x<x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.5782em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>&lt; is transitive:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x<y, y<z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x<z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span>.</p>
+<p><strong>&lt; is transitive </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x<y, y<z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x<z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span>.</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Trichotomy: </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x=y or (x<y or y<x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7335400000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Trichotomy </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x=y or (x<y or y<x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.7335400000000001em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 </div>
+<div>&nbsp;</div>
+<div>The following axiom is not used in our course but we list it here for completeness (pun).</div>
+<div>&nbsp;</div>
 <div><strong>&nbsp; &nbsp; Completion:</strong> Every bounded above sequence of real numbers has a least upper bound.</div>
-<div>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(<em>not used in our course</em>)</div>
+<div>&nbsp;</div>
 <h4>Axioms Relating +, ⋅, and &lt;</h4>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Distributivity: </strong>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x\\cdot (y+z) = x\\cdot y + x\\cdot z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span></p>
+<p><strong>Distributivity </strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expositorymath&quot;" data-metadata_latex="&quot;x\\cdot (y+z) = x\\cdot y + x\\cdot z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__cmr">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">)</span><span class="ML__cmr">=</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Translation:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x<y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x+z<y+z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span></p>
+<p><strong>Translation</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x<y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x+z<y+z&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">+</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Product of positives:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;if 0<x, 0<y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">if </span><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__text"> and </span><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0<x*y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span></p>
+<p><strong>Product of positives </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If 0<x, 0<y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__text"> and </span><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;0<x*y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr">&lt;</span><span class="ML__mathit">x</span><span class="ML__cmr">⋅</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span></p>
 </div>
 </div>
 <hr>
 <h3>&nbsp;</h3>
-</div>
 </div></div>
         </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Reals-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div></div>
             <div id="document"><h1>Set Theory</h1>
@@ -634,100 +770,121 @@
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Declare in, subset, cap, cup, complement, setminus , powerset, cross, set, tuple, Union, Intersect, setbuilder&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">∈</span><span class="ML__text">, </span><span class="ML__cmr">⊆</span><span class="ML__text">, </span><span class="ML__cmr">∩</span><span class="ML__text">, </span><span class="ML__cmr">∪</span><span class="ML__text">, </span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.81em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.39em; display: inline-block; font-size: 70%;"><span class="ML__cmr">′</span></span></span></span></span></span></span><span class="ML__text">, </span><span class="ML__cmr">∖</span><span class="ML__text">, </span><span class="ML__script" style="margin-right: 0.09em;">P</span><span class="ML__text">, </span><span class="ML__cmr">×</span><span class="ML__text">, set, tuple, Union, Intersect, and setbuilder</span></span></span></span></span> to be constants.</p>
 <h3>Definitions</h3>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Empty set: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x notin Set()&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__left-right" style="margin-top: 0em; height: 0em;"><span class="ML__small-delim ML__open">{</span><span class="ML__small-delim ML__close">}</span></span></span></span></span></span></p>
+<p><strong>Empty set&nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x notin Set()&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__left-right" style="margin-top: 0em; height: 0em;"><span class="ML__small-delim ML__open">{</span><span class="ML__small-delim ML__close">}</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Nonempty: </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A neq { }&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: 0em; height: 0em;"><span class="ML__small-delim ML__open">{</span><span class="ML__small-delim ML__close">}</span></span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists x.x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
+<p><strong>Nonempty </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A neq { }&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__rlap"><span class="ML__inner"><span class="ML__cmr"></span></span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: 0em; height: 0em;"><span class="ML__small-delim ML__open">{</span><span class="ML__small-delim ML__close">}</span></span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;exists x.x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">∃</span><span class="ML__mathit">x</span><span class="ML__cmr">.</span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
 <p><strong>Finite set notation:</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a) equiv x=a&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a) equiv x=a&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b) equiv x=a or x=b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b) equiv x=a or x=b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b,c) equiv x=a or (x=b or x=c)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b,c) equiv x=a or (x=b or x=c)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b,c,d) equiv \nx=a or (x=b or (x=c or x=d))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__cmr">,</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in set(a,b,c,d) equiv \nx=a or (x=b or (x=c or x=d))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__cmr">,</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">}</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">a</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">b</span><span class="ML__text"> or </span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Set builder: &nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in { z : @P(z) } equiv @P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">:</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">}</span></span><span class="ML__text"> iff </span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Set builder&nbsp;</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in { z : @P(z) } equiv @P(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">{</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">:</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">}</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__cal" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Subset-:&nbsp;</strong><br><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;From x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">From </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Subset-:&nbsp;</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Subset+</strong>:</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+</div>
+<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Set equality:</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
-<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
+<p>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A=B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+</div>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A=B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span><strong>&nbsp;</strong></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Set equality:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If A=B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">A</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;B subset A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">⊆</span><span class="ML__mathit">A</span></span></span></span></span></p>
+<p><strong>Set equality</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A=B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> ,&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;B subset A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">⊆</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Powerset:&nbsp;&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A in powerset(B) equiv A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">∈</span><span class="ML__script" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> iff </span><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Powerset&nbsp; </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;A in powerset(B) equiv A subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">∈</span><span class="ML__script" style="margin-right: 0.09em;">P</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">A</span><span class="ML__cmr">⊆</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Intersection:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A intersect B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∩</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span> <em>and</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Intersection </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A intersect B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∩</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span> , <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Union:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in A union B equiv x in A or x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∪</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Union</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in A union B equiv x in A or x in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∪</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">&nbsp;</span><span class="ML__cmr">≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Relative Complement:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A setminus B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∖</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span>&nbsp;<em>and</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x notin B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Relative Complement </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A setminus B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">∖</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span> , <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x notin B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Complement:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B complement equiv x notin B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.81em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.39em; display: inline-block; font-size: 70%;"><span class="ML__cmr">′</span></span></span></span></span></span></span><span class="ML__text"> iff </span><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><strong>Complement </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in B complement equiv x notin B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.81em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.39em; display: inline-block; font-size: 70%;"><span class="ML__cmr">′</span></span></span></span></span></span></span><span class="ML__cmr">&nbsp;≡&nbsp;</span><span class="ML__mathit">x</span><span class="ML__cmr">∉</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <p><strong>Ordered tuple:</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>(pair):&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;⟨a,b⟩=⟨c,d⟩&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">c</span><span class="ML__cmr">,</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">⟩</span></span></span></span></span></span>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a=c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span></span></span></span></span>&nbsp;<em>and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=d&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span></span></span></span></span></em></p>
+<p><strong>(pair) </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;⟨a,b⟩=⟨c,d⟩&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">c</span><span class="ML__cmr">,</span><span class="ML__mathit">d</span><span class="ML__small-delim ML__close">⟩</span></span></span></span></span></span>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a=c&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">=</span><span class="ML__mathit">c</span></span></span></span></span> ,<em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=d&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span></span></span></span></span></em></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>(triple):&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;⟨a,b,c⟩=⟨d,e,f⟩&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">d</span><span class="ML__cmr">,</span><span class="ML__mathit">e</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__small-delim ML__close">⟩</span></span></span></span></span></span>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a=d&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span></span></span></span></span>,<em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=e&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit">e</span></span></span></span></span>, and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;c=f&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">c</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span></span></span></span></span></em></p>
+<p><strong>(triple) </strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;⟨a,b,c⟩=⟨d,e,f⟩&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">d</span><span class="ML__cmr">,</span><span class="ML__mathit">e</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__small-delim ML__close">⟩</span></span></span></span></span></span>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;≡&nbsp;</span></span></span></span></span> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a=d&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">=</span><span class="ML__mathit">d</span></span></span></span></span>,<em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=e&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit">e</span></span></span></span></span>, <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;c=f&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">c</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span></span></span></span></span></em></p>
 </div>
 </div>
 <div>
 <p><strong>Cartesian product</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>(pair):&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If z in A cross B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">×</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z=⟨a,b⟩ and (a in A and b in B) for some a,b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__text"> and </span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.73354em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__text"> and </span><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">a</span><span class="ML__text"> and </span><span class="ML__mathit">b</span></span></span></span></span>.</p>
+<p><strong>(pair): </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If z in A cross B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">×</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;z=⟨a,b⟩ and (a in A and b in B) for some a,b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.05em;">z</span><span class="ML__cmr">=</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__text"> and </span><span class="ML__left-right" style="margin-top: -0.0391em; height: 0.73354em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__text"> and </span><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">a</span><span class="ML__text"> and </span><span class="ML__mathit">b</span></span></span></span></span>.</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>(pair):&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If pair(a,b) in A cross B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">×</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span>.</p>
@@ -747,29 +904,29 @@
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>(triple):</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If triple(a,b,c) in A cross B cross C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">⟨</span><span class="ML__mathit">a</span><span class="ML__cmr">,</span><span class="ML__mathit">b</span><span class="ML__cmr">,</span><span class="ML__mathit">c</span><span class="ML__small-delim ML__close">⟩</span></span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span><span class="ML__cmr">×</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">×</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">∈</span><span class="ML__mathit">A</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;c in C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">c</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span>.</p>
 </div>
-<hr>
-<h4>Indexed Union&nbsp;</h4>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x in Union i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋃</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in @A(j) for some j in I&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span></span></span>.</p>
+<p><strong>Indexed Union: </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x in Union i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋃</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in @A(j) for some j in I&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span></span></span>.</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If j in I, x in @A(j) &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span><span class="ML__text"> and </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in Union i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋃</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+<p><strong>Indexed Union: </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If j in I, x in @A(j) &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span><span class="ML__text"> and </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in Union i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋃</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<h4>Indexed Intersection</h4>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x in Intersect i in I of @A(i), j in I&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋂</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in @A(j)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>.</p>
+<p><strong>Indexed Intersection:&nbsp; </strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x in Intersect i in I of @A(i), j in I&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋂</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in @A(j)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>.</p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p>From the subproof:</p>
+<p><strong>Indexed Intersection</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let j in I&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in @A(j)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.85396em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">j</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p>we can conclude <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in Intersect i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋂</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>.</p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in Intersect i in I of @A(i)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">∈</span><span class="ML__op-group"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 1.06em;"><span class="ML__center" style="top: -1.85em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__mathit">i</span><span class="ML__cmr">∈</span><span class="ML__mathit" style="margin-right: 0.08em;">I</span></span></span><span class="ML__center" style="top: -3.05em;"><span style="height: 1.61em; display: inline-block;"><span class="ML__op-symbol ML__large-op">⋂</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cal">A</span><span class="ML__left-right" style="margin-top: 0em; height: 0.65952em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">i</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>.</p>
 </div>
 <hr></div>
 </div></div>
-        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Sets-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div><div data-category="settings" data-key="shell style" data-value-type="json">"minimal"</div></div>
+        </div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Source:</td><td><tt>math/Sets-Rules.lurch</tt> (web)</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div><div data-category="settings" data-key="shell style" data-value-type="json">"boxed"</div></div>
             <div id="document"><div>
 <h1>Functions</h1>
 <p>We now define a formal system for defining elementary concepts related to functions.&nbsp; This library includes all of the rules, definitions, and axioms from the Sets library.</p>
@@ -789,18 +946,22 @@
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Function Equality:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, g:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
-<p>and we have the subproof:</p>
+<p><strong>Function Equality</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;g:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
 <div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let x in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">x</span><span class="ML__cmr">&isin;</span><span class="ML__mathit">A</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(x)=g(x)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p>then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f=g&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span></span></span></span></span>.</p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f=g&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span></span></span></span></span>.</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Image of a Set:</strong> If <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x in S&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span> then &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(x) in f(S)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>.</p>
+<p><strong>Image of a Set:</strong>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If x in S&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit">x</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span> then&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(x) in f(S)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Image of a Set:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;if y in f(S)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">if </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;y=f(x) for some x in S&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">x</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span>.</p>
@@ -808,79 +969,133 @@
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Identity map:</strong>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;id_(A):A to A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">id</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.33em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.48em; display: inline-block; font-size: 70%;"><span class="ML__mathit">A</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit">A</span></span></span></span></span>.</p>
+<p><strong>Identity map</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;id_(A):A to A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">id</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.33em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.48em; display: inline-block; font-size: 70%;"><span class="ML__mathit">A</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Identity map:</strong>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;id_(A)(x)=x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">id</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.33em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.48em; display: inline-block; font-size: 70%;"><span class="ML__mathit">A</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">x</span></span></span></span></span>.</p>
+<p><strong>Identity map</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;id_(A)(x)=x&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">id</span><span class="ML__msubsup"><span class="ML__vlist-t ML__vlist-t2"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.33em;"><span style="top: -2.85em; margin-right: 0.05em;"><span style="height: 0.48em; display: inline-block; font-size: 70%;"><span class="ML__mathit">A</span></span></span></span><span class="ML__vlist-s">​</span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">x</span></span></span></span></span>.</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Composition:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, g:B to C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">:</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;g comp f:A to C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">∘</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span></p>
+<p><strong>Composition</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;g:B to C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">:</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span></p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;g comp f:A to C&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">∘</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.08em;">C</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Composition:&nbsp;</strong><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(g comp f)(x)=g(f(x))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">∘</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
+<p><strong>Composition</strong></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;(g comp f)(x)=g(f(x))&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8888799999999999em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__cmr">∘</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">g</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp;</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Injective:</strong> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f is injective, f(x)=f(y)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective and </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x=y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span>.</p>
+<p><strong>Injective</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is injective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective</span></span></span></span></span></p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(x)=f(y)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x=y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Injective:</strong>&nbsp; From the subproof&nbsp;</p>
+<p><strong>Injective:</strong></p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let x,y be such that f(x)=f(y)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">x</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__text"> be such that </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">x</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.625em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;x=y&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">x</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.04em;">y</span></span></span></span></span></p>
 </div>
-<p>we can conclude that <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is injective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective</span></span></span></span></span>.</p>
+</div>
+<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is injective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Surjective:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, f is surjective, b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text">, </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective, and </span><span class="ML__mathit">b</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=f(a) for some a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">a</span><span class="ML__cmr">&isin;</span><span class="ML__mathit">A</span></span></span></span></span>.</p>
+<p><strong>Surjective</strong>&nbsp;</p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is surjective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude</em>&nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=f(a) for some a in A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> for some </span><span class="ML__mathit">a</span><span class="ML__cmr">&isin;</span><span class="ML__mathit">A</span></span></span></span></span>.</p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
 <p><strong>Surjective:</strong>&nbsp;</p>
-<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> and we have the subproof</p>
 <div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<div class="lurch-atom" data-metadata_type="&quot;subproof&quot;" data-shell_title="">
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;Let b in B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Let </span><span class="ML__mathit">b</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></p>
 <p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;b=f(a)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">b</span><span class="ML__cmr">=</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
 </div>
-<p>then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is surjective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective</span></span></span></span></span>.</p>
+</div>
+<p><span style="background-color: var(--semantic-hover-color); font-size: var(--base-font-size);"><em>Conclude</em> </span><span class="lurch-atom" style="font-size: var(--base-font-size);" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is surjective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Bijective:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is injective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective</span></span></span></span></span> and <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is surjective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective</span></span></span></span></span></p>
+<p><strong>Bijective</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;&equiv;&nbsp;</span></span></span></span></span> &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is injective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is injective</span></span></span></span></span> ,&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is surjective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is surjective</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Inverse map:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span> then<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^(-1):B to A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__cmr">:</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit">A</span></span></span></span></span></p>
+<p><strong>Inverse map</strong>&nbsp;</p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span></em></p>
+</div>
+<p><em>&nbsp;Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^(-1):B to A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__cmr">:</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Inverse map:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, f is bijective &quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^(-1)(f(a))=a&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">a</span></span></span></span></span>.</p>
+<p><strong>Inverse map</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^(-1)(f(a))=a&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: -0.25em; height: 1em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">a</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Inverse map:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f:A to B, f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(f^(-1)(b))=b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1.1141079999999999em;"><span class="ML__open ML__delim-size1">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__close ML__delim-size1">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">b</span></span></span></span></span>.</p>
+<p><strong>Inverse map</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f:A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f is bijective&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__text"> is bijective</span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude</em>&nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(f^(-1)(b))=b&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: -0.25em; height: 1.1141079999999999em;"><span class="ML__open ML__delim-size1">(</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.87em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.51em; display: inline-block; font-size: 70%;"><span class="ML__cmr">&minus;1</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.69444em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">b</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__close ML__delim-size1">)</span></span><span class="ML__cmr">=</span><span class="ML__mathit">b</span></span></span></span></span></p>
 </div>
 </div>
 <div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Inverse image:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in f^inv(S)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.89em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__text">inv</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text"> iff </span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(a) in S&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span></p>
+<p><strong>Inverse image</strong>&nbsp;</p>
+<p><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;a in f^inv(S)&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">a</span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.89em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__text">inv</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span>&nbsp; &nbsp;<span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;equiv&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">&nbsp;&equiv;&nbsp;</span></span></span></span></span> &nbsp; <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f(a) in S&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__left-right" style="margin-top: 0em; height: 0.43056em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">a</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&isin;</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span></p>
 </div>
 <div class="lurch-atom" data-metadata_type="&quot;rule&quot;" data-shell_title="Rule:">
-<p><strong>Inverse image:</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;If f: A to B, U subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">If </span><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__cmr">&sube;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span> then <span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^inv(U) subset A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.89em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__text">inv</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&sube;</span><span class="ML__mathit">A</span></span></span></span></span></p>
+<p><strong>Inverse image</strong></p>
+<div class="lurch-atom" data-metadata_type="&quot;premise&quot;" data-shell_title="">
+<p><em>Given</em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f: A to B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__cmr">:</span><span class="ML__mathit">A</span><span class="ML__cmr">&rarr;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;U subset B&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__cmr">&sube;</span><span class="ML__mathit" style="margin-right: 0.06em;">B</span></span></span></span></span></em></p>
+</div>
+<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type="&quot;expression&quot;" data-metadata_lurch-notation="&quot;f^inv(U) subset A&quot;"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">f</span><span class="ML__msubsup"><span class="ML__vlist-t"><span class="ML__vlist-r"><span class="ML__vlist" style="height: 0.89em;"><span style="top: -3.41em; margin-right: 0.05em;"><span style="height: 0.47em; display: inline-block; font-size: 70%;"><span class="ML__text">inv</span></span></span></span></span></span></span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">&sube;</span><span class="ML__mathit">A</span></span></span></span></span></p>
 </div>
 </div>
 <div>&nbsp;</div></div>