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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>, <font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:78"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font>
 ;<br/>

<b>let </b><font color="Maroon">u1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">u2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>;<br/>



<b>assume </b><a NAME="E2:78"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font>
 ;<br/>

<b>assume </b><a NAME="E3:78"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u1</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_1"/>
( <font color="Maroon">G1</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">u1</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">G1</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">u1</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="78_1_1_1"><b>suppose </b></a><a NAME="E1:78_1_1_1"/>
<font color="Maroon">G1</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">u1</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"><a NAME="E2:78_1_1_1"/><b>then </b>
<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">u2</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#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E3:78_1_1_1"/>
<font color="Maroon">u2</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_2"><b>suppose </b></a><a NAME="E1:78_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">G1</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">u1</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"><b>consider </b><font color="Maroon">H1</font> being    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G1</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u1</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E3:78_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">H1</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#E4:78_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="chord.html#E3:78_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H2</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">u2</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E1:78_1_1_2_1"/>
<font color="Maroon">H1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">H2</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E3:78"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E2:78_1_1_2_1"/>
<font color="Maroon">H2</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E5:78_1_1_2"/>
<font color="Maroon">u2</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E3:78_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:78"/>
<font color="Maroon">u2</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>


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