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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="chord.html#V6" target="_self">chordal</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:122"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">G</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="glib_000.html#V4">trivial</a> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="chord.html#V6" target="_self">chordal</a> &amp; not <font color="Maroon">a<sub>1</sub></font> is <a href="chord.html#V2" target="_self">complete</a> implies  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">a<sub>1</sub></font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> &amp; <font color="Olive">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> ) );<br/>
<a NAME="E2:122"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>  st ( for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Gk</font>] ) holds <br/>for <font color="Olive">Gk1</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Gk1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Gk</font>]
 ;<br/>

<b>let </b><font color="Maroon">Gk1</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:122_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E3:122_1"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">Gk1</font> is <a href="glib_000.html#V4">trivial</a>
 <b>and </b><br/><a NAME="E4:122_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">Gk1</font> is <a href="chord.html#V6" target="_self">chordal</a>
 <b>and </b><br/><a NAME="E5:122_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">Gk1</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">Gk1</font> as  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="chord.html#V6" target="_self">chordal</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:122_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:122_1"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E11:122_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Gs</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/>
<b>consider </b><font color="Maroon">Gns</font> being   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/>
<b>reconsider </b><font color="Maroon">sa</font> = <font color="Maroon">a</font>, <font color="Maroon">sb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gns</font> <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="txt" href="chord.html#E20:122_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>;<br/>
<b>consider </b><font color="Maroon">Gas</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">Gbs</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E17:122_1"/>
( <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</font> &amp; <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</font> )
 
;<br/>
<a NAME="E18:122_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:122_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font> is  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> &amp; <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font> is  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> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E20:122_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="txt" href="chord.html#E19:122_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:122_1_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="txt" href="chord.html#E4:122_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E3:122_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:122_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/><b>hence </b><a NAME="E5:122_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:122_1_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="txt" href="chord.html#E4:122_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E3:122_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:122_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9">GLIB_002:9</a></i>;<br/><b>hence </b><a NAME="E5:122_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:122_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="txt" href="chord.html#E4:122_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E24:122_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E25:122_1"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E26:122_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Gas</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T78">GLIB_000:78</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Gas</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:122_1_3"/><i><font color="Green">E86</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gas</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/><a NAME="E3:122_1_3"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gas</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E4:122_1_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E28:122_1"><i><font color="Green">E112</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:122_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">Gas</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E29:122_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">Gbs</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T78">GLIB_000:78</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">Gbs</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:122_1_4"/><i><font color="Green">E115</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gbs</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E29:122_1"><i><font color="Green">E113</font></i></a>, , <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/><a NAME="E3:122_1_4"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gbs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E4:122_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="txt" href="chord.html#E1:122_1_4"><i><font color="Green">E114</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:122_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">Gbs</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E28:122_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E32:122_1"/>
 ex <font color="Olive">a</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Olive">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:122_1_5"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:122_1_5"/><i><font color="Green">E119</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gas</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">aa</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:122_1_5"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Olive">c</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_5_1_1"/>
( <font color="Maroon">Gas</font> is <a href="chord.html#V2" target="_self">complete</a> or  not <font color="Maroon">Gas</font> is <a href="chord.html#V2" target="_self">complete</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:122_1_5_1_1_1"/>
<font color="Maroon">Gas</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_5_1_1_1"/><b>then </b>
<font color="Maroon">a</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:122_1_5_1_1_1"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Olive">c</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="122_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:122_1_5_1_1_2"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">Gas</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_5_1_1_2"/>
not <font color="Maroon">Gas</font> is <a href="glib_000.html#V4">trivial</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E2:122_1"><i><font color="Green">E56</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">a2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font>, <font color="Maroon">b2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font><b> such that </b><br/><a NAME="E4:122_1_5_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E5:122_1_5_1_1_2"/><i><font color="Green">E123</font></i>: 
not <font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>and </b><br/><a NAME="E6:122_1_5_1_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">a2</font> is <a href="chord.html#V3" target="_self">simplicial</a> &amp; <font color="Maroon">b2</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_5_1_1_2_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a3</font> = <font color="Maroon">a2</font>, <font color="Maroon">b3</font> = <font color="Maroon">b2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E3:122_1_5_1_1_2_1"/><i><font color="Green">E126</font></i>: 
not <font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green">E118</font></i></a>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:122_1_5_1_1_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">Gs</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T19" target="_self">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon">a4</font> = <font color="Maroon">a2</font>, <font color="Maroon">b4</font> = <font color="Maroon">b2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gs</font> <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:122_1_5_1_1_2_1"/>
<font color="Maroon">a4</font>,<font color="Maroon">b4</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:122_1_5_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, </i>;<br/><b>hence </b><a NAME="E7:122_1_5_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><a NAME="E8:122_1_5_1_1_2"/><b>then </b>
( <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> or <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:122_1_5_1_1_2"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Olive">c</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_5"><i><font color="Green">E119</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">cc</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gas</font><b> such that </b><br/><a NAME="E7:122_1_5"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 <b>and </b><br/><a NAME="E8:122_1_5"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">cc</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">cc</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:122_1_5_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gk1</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">c</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>assume </b><a NAME="E2:122_1_5_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E3:122_1_5_2"/><b>then </b><i><font color="Green">E198</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> <b>by </b><i><a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green">E118</font></i></a></i>;<br/><a NAME="E5:122_1_5_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E3:122_1_5"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:122_1_5_2"/>
<font color="Maroon">c</font>,<font color="Maroon">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green">E118</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></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="E8:122_1_5_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E9:122_1_5_2"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">Gns</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T27" target="_self">Th27</a>, , , </i>;<br/><b>hence </b><a NAME="E10:122_1_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_5"><i><font color="Green">E119</font></i></a>, , <a class="ref" href="glib_002.html#T10">GLIB_002:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:122_1_5"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">Gk1</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">c</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:122_1_5"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:122_1_5"/><b>then </b>
<font color="Maroon">c</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 
<b>by </b><i>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green">E118</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E14:122_1_5"/>
 ex <font color="Olive">a</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Olive">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">a</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font><b> such that </b><br/><a NAME="E34:122_1"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font> &amp; <font color="Maroon">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 ;<br/>
<a NAME="E35:122_1"/>
 ex <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Olive">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:122_1_6"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:122_1_6"/><i><font color="Green">E204</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gbs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">aa</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:122_1_6"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Olive">c</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_6_1_1"/>
( <font color="Maroon">Gbs</font> is <a href="chord.html#V2" target="_self">complete</a> or  not <font color="Maroon">Gbs</font> is <a href="chord.html#V2" target="_self">complete</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:122_1_6_1_1_1"/>
<font color="Maroon">Gbs</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_6_1_1_1"/><b>then </b>
<font color="Maroon">a</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:122_1_6_1_1_1"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Olive">c</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="122_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:122_1_6_1_1_2"/><i><font color="Green">E205</font></i>: 
not <font color="Maroon">Gbs</font> is <a href="chord.html#V2" target="_self">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_6_1_1_2"/>
not <font color="Maroon">Gbs</font> is <a href="glib_000.html#V4">trivial</a>
 <b>by </b><i><a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E2:122_1"><i><font color="Green">E56</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">a2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font>, <font color="Maroon">b2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font><b> such that </b><br/><a NAME="E4:122_1_6_1_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E5:122_1_6_1_1_2"/><i><font color="Green">E207</font></i>: 
not <font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>and </b><br/><a NAME="E6:122_1_6_1_1_2"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">a2</font> is <a href="chord.html#V3" target="_self">simplicial</a> &amp; <font color="Maroon">b2</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="chord.html#E29:122_1"><i><font color="Green">E113</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_6_1_1_2_1"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a3</font> = <font color="Maroon">a2</font>, <font color="Maroon">b3</font> = <font color="Maroon">b2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E3:122_1_6_1_1_2_1"/><i><font color="Green">E210</font></i>: 
not <font color="Maroon">a3</font>,<font color="Maroon">b3</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:122_1_6_1_1_2_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">Gs</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T19" target="_self">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon">a4</font> = <font color="Maroon">a2</font>, <font color="Maroon">b4</font> = <font color="Maroon">b2</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gs</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:122_1_6_1_1_2_1"/>
<font color="Maroon">a4</font>,<font color="Maroon">b4</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:122_1_6_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><a NAME="E8:122_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> or <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:122_1_6_1_1_2"/>
 ex <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Olive">c</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">cc</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gbs</font><b> such that </b><br/><a NAME="E7:122_1_6"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font>
 <b>and </b><br/><a NAME="E8:122_1_6"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">cc</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">cc</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> <b>by </b><i>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:122_1_6_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gk1</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">c</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>assume </b><a NAME="E2:122_1_6_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E3:122_1_6_2"/><b>then </b><i><font color="Green">E215</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> <b>by </b><i/>;<br/><a NAME="E5:122_1_6_2"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gns</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:122_1_6_2"/>
<font color="Maroon">c</font>,<font color="Maroon">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></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="E8:122_1_6_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E9:122_1_6_2"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>,<font color="Maroon">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, , , , </i>;<br/><b>hence </b><a NAME="E10:122_1_6_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T27" target="_self">Th27</a>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="glib_002.html#T10">GLIB_002:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:122_1_6"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">Gk1</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">c</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:122_1_6"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:122_1_6"/><b>then </b>
<font color="Maroon">c</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E14:122_1_6"/>
 ex <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Olive">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">b</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font><b> such that </b><br/><a NAME="E37:122_1"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sb</font> &amp; <font color="Maroon">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 ;<br/>
<div><i><font color="Green">E220</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_7"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<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="E3:122_1_7"/><i><font color="Green">E221</font></i>: 
<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">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font>, <font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gns</font> <b>by </b><i>, </i>;<br/><a NAME="E5:122_1_7"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">aa</font>,<font color="Maroon">bb</font>,<font color="Maroon">Gns</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green">E117</font></i></a>, </i>;<br/><a NAME="E6:122_1_7"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gns</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">sa</font>
 <b>by </b><i>, , <a class="ref" href="glib_002.html#T10">GLIB_002:10</a></i>;<br/><b>hence </b><a NAME="E7:122_1_7"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E39:122_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E40:122_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> &amp; <font color="Olive">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:122"/>
for <font color="Olive">G</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>]
 
<b>from </b><i><a class="ref" href="chord.html#S1" target="_self">CHORD:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E4:122"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font> is <a href="chord.html#V3" target="_self">simplicial</a> &amp; <font color="Olive">b</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 <b>by </b><i/>;<br/>


</div>
