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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E2:79"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E3:79"/><i><font color="Green">E53</font></i>: 
( <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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Maroon">G2</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">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="chord.html#T16" target="_self">Th16</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1"/>
( <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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1"><b>suppose </b></a><a NAME="E1:79_1_1"/>
<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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_1_1"/>
( <font color="Maroon">u</font> is <a href="chord.html#V3" target="_self">simplicial</a> iff <font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2"><b>suppose </b></a><a NAME="E1:79_1_2"/><i><font color="Green">E56</font></i>: 
<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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:79_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G2</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">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T16" target="_self">Th16</a>, , </i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:79_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">u</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/><b>consider </b><font color="Maroon">Ga</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">u</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E3:79_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">Ga</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i><a class="txt" href="chord.html#E6:79_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chord.html#E3:79_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="chord.html#E2:79_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>thus </b><a NAME="E4:79_1_2_1"/>
<font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1_2_1_1"/>
<font color="Maroon">G2</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">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D7" target="_self">CHORD:def 7</a><br/>

<b>let </b><font color="Maroon">Ha</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G2</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/>

<a NAME="E2:79_1_2_1_1"/>
<font color="Maroon">Ga</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">Ha</font>
 
<b>by </b><i>, <a class="ref" href="chord.html#T16" target="_self">Th16</a>, <a class="txt" href="chord.html#E3:79_1_2_1"><i><font color="Green">E67</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:79_1_2_1_1"/>
<font color="Maroon">Ha</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E4:79_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/><b>consider </b><font color="Maroon">Ha</font> being    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G2</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E6:79_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Ha</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i><a class="txt" href="chord.html#E6:79_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chord.html#E4:79_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chord.html#E2:79_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>thus </b><a NAME="E7:79_1_2"/>
<font color="Maroon">u</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1_2_2"/>
<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">u</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D7" target="_self">CHORD:def 7</a><br/>

<b>let </b><font color="Maroon">Ga</font> be    <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">u</font></span><a href="domain_1.html#K6">}</a></span>;<br/>

<a NAME="E2:79_1_2_2"/>
<font color="Maroon">Ga</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">Ha</font>
 
<b>by </b><i>, <a class="ref" href="chord.html#T16" target="_self">Th16</a>, , </i>;<br/>
<b>hence </b><a NAME="E3:79_1_2_2"/>
<font color="Maroon">Ga</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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