<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:80"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:80"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<b>consider </b><font color="Maroon">H</font> being    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E4:80"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">H</font> is <a href="chord.html#V2" target="_self">complete</a>
 
<b>by </b><i><a class="ref" href="chord.html#T16" target="_self">Th16</a>, , <a class="txt" href="chord.html#E10:80"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">H</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">a</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font> <b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font> <b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E8:80"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:80"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E11:80"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/>
<b>hence </b><a NAME="E12:80"/>
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">G</font>
 ;<br/>


</div>
