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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span><br/> for <font color="Olive" title="b2">H</font> being  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">u</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> holds <br/>for <font color="Olive" title="b4">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">H</font>  st <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> holds <br/>( <font color="Olive" title="b3">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Olive" title="b4">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">u</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b3">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">H</font>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> holds <br/>( <font color="Olive" title="b2">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Olive" title="b3">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/><b>let </b><font color="Maroon" title="c3">G2</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b2">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/>( <font color="Olive" title="b1">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Olive" title="b2">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/><b>let </b><font color="Maroon" title="c4">u</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> implies for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>  st <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/>( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Olive" title="b1">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> ) )</span><br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E64">A1</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>  st <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/>( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Olive" title="b1">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/>

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> implies ( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> ) )</span><br/>

<b>assume </b><a NAME="E2:79"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/>

<a NAME="E3:79"/><i><font color="Green" title="E66">A3</font></i>: 
( <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c3">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="chord.html#T58" target="_self" title="CHORD:th.58">Th58</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" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/><div class="add"><b>hence </b><a NAME="E2:79_1_1"/>
( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="#E3:79"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="chord.html#D7" target="_self" title="CHORD:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E67">A4</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> iff <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span><br/><div class="add"><a NAME="E2:79_1_2"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c3">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="chord.html#T58" target="_self" title="CHORD:th.58">Th58</a></i>;<br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> implies <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:79_1_2_1"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a></span><br/><b>consider </b><font color="Maroon" title="c6">Ga</font> being    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E3:79_1_2_1"/><i><font color="Green" title="E70">A7</font></i>: 
<font color="Maroon" title="c6">Ga</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="#E1:79_1_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="#E1:79_1_2_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="chord.html#D7" target="_self" title="CHORD:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E4:79_1_2_1"/>
<font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c3">G2</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D7" target="_self" title="CHORD:def.7">CHORD:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being   <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c3">G2</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Olive" title="b1">G2</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a></span><br/>

<b>let </b><font color="Maroon" title="c7">Ha</font> be    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c3">G2</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ha</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a></span><br/>

<b>thus </b><a NAME="E2:79_1_2_1_1"/>
<font color="Maroon" title="c7">Ha</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E1:79_1_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="#E3:79_1_2_1"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="chord.html#T60" target="_self" title="CHORD:th.60">Th60</a>, <a class="ref" href="chord.html#T62" target="_self" title="CHORD:th.62">Th62</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E4:79_1_2"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a></span><br/><b>consider </b><font color="Maroon" title="c6">Ha</font> being    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c3">G2</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E6:79_1_2"/><i><font color="Green" title="E70">A9</font></i>: 
<font color="Maroon" title="c6">Ha</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="#E2:79_1_2"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="#E4:79_1_2"><i><font color="Green" title="E69">A8</font></i></a>, <a class="ref" href="chord.html#D7" target="_self" title="CHORD:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E7:79_1_2"/>
<font color="Maroon" title="c4">u</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D7" target="_self" title="CHORD:def.7">CHORD:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being   <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Olive" title="b1">G2</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a></span><br/>

<b>let </b><font color="Maroon" title="c7">Ga</font> be    <a href="chord.html#M1" title="CHORD:mode.1">AdjGraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ga</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a></span><br/>

<a NAME="E2:79_1_2_2"/>
<font color="Maroon" title="c7">Ga</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c6">Ha</font>
 
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E1:79_1_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="chord.html#T60" target="_self" title="CHORD:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E3:79_1_2_2"/>
<font color="Maroon" title="c7">Ga</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="#E6:79_1_2"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="chord.html#T62" target="_self" title="CHORD:th.62">Th62</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

</div>
