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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>] <span class="kw">means </span>(  not $1 is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  &amp; $1 is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  &amp;  not $1 is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of $1 st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  ) );<br/>
<a NAME="E1:121"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st (  for <font color="Olive" title="b2">Gk</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b2">Gk</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">Gk</font>] ) holds <br/> for <font color="Olive" title="b2">Gk1</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b2">Gk1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">Gk1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">Gk</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">Gk</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk</font>] ) implies  for <font color="Olive" title="b1">Gk1</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">Gk1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk1</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:121_1"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
 for <font color="Olive" title="b1">Gk</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">Gk</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Gk1</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">Gk1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk1</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">Gk1</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">Gk1</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:121_1"/><span class="lab"><font color="Green" title="E95">A3</font></span>: 
<font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E3:121_1"/><span class="lab"><font color="Green" title="E96">A4</font></span>: 
 not <font color="Maroon" title="c2">Gk1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E4:121_1"/><span class="lab"><font color="Green" title="E97">A5</font></span>: 
<font color="Maroon" title="c2">Gk1</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">and </span><br/><a NAME="E5:121_1"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
 not <font color="Maroon" title="c2">Gk1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">G</font> = <font color="Maroon" title="c2">Gk1</font> as   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>   <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_1"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="txt" href="#E4:121_1"><span class="lab"><font color="Green" title="E97">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E8:121_1"/><span class="lab"><font color="Green" title="E99">A7</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E9:121_1"/><span class="lab"><font color="Green" title="E100">A8</font></span>: 
 not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121_1"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font><span class="kw"> such that </span><br/><a NAME="E11:121_1"/><span class="lab"><font color="Green" title="E101">A9</font></span>: 
<font color="Maroon" title="c6">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T79" target="_self" title="CHORD:th.79">Th79</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Gns</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">sa</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c9">sb</font> = <font color="Maroon" title="c5">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="chord.html#T77" target="_self" title="CHORD:th.77">Th77</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">B</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Gas</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span>;<br/>
<a NAME="E13:121_1"/><span class="lab"><font color="Green" title="E102">A10</font></span>: 
 the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 
;<br/>
<a NAME="E14:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A11</font></span>: 
 the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E15:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A12</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font> is   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="c3">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E16:121_1"/><span class="lab"><font color="Green" title="E105">A13</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="chord.html#T76" target="_self" title="CHORD:th.76">Th76</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E106">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_1_1"/><span class="lab"><font color="Green" title="E107">A15</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:121_1_1"/>
 not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>;<br/><a NAME="E3:121_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A16</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_1"><span class="lab"><font color="Green" title="E107">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:121_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:121_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:121_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="txt" href="#E3:121_1_1"><span class="lab"><font color="Green" title="E108">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E107">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_1_2"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:121_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A18</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="ref" href="pre_poly.html#T8" title="PRE_POLY:th.8">PRE_POLY:8</a></span>;<br/><a NAME="E3:121_1_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:121_1"><span class="lab"><font color="Green" title="E104">A12</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">hence </span><a NAME="E4:121_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:121_1"><span class="lab"><font color="Green" title="E106">A14</font></span></a>, <a class="txt" href="#E2:121_1_2"><span class="lab"><font color="Green" title="E108">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c13">Gbs</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span>;<br/>
<a NAME="E19:121_1"/><span class="lab"><font color="Green" title="E108">A19</font></span>: 
 the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:121_1"><span class="lab"><font color="Green" title="E102">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A20</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font> is   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="c3">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E110">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_1_3"/><span class="lab"><font color="Green" title="E111">A22</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:121_1_3"/>
 not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>;<br/><a NAME="E3:121_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A23</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_3"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:121_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:121_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:121_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="txt" href="#E3:121_1_3"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E111">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_1_4"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:121_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A25</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="ref" href="pre_poly.html#T8" title="PRE_POLY:th.8">PRE_POLY:8</a></span>;<br/><a NAME="E3:121_1_4"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:121_1"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">hence </span><a NAME="E4:121_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:121_1"><span class="lab"><font color="Green" title="E110">A21</font></span></a>, <a class="txt" href="#E2:121_1_4"><span class="lab"><font color="Green" title="E112">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c14">Gs</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>;<br/>
<a NAME="E23:121_1"/>
 not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>;<br/>
<a NAME="E24:121_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c3">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E25:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A26</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E26:121_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></span>;<br/>
<a NAME="E27:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A27</font></span>: 
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:121_1"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E28:121_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c15">aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:121_1_5"/><span class="lab"><font color="Green" title="E114">A28</font></span>: 
<font color="Maroon" title="c15">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E3:121_1_5"/><span class="lab"><font color="Green" title="E115">A29</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:121_1"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c15">aa</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_5"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:121_1_5"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:121_1_5_1_1"/>
(  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  or  not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:121_1_5_1_1_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/><div class="add"><a NAME="E2:121_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T64" target="_self" title="CHORD:th.64">Th64</a></span>;<br/><span class="kw">hence </span><a NAME="E3:121_1_5_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_5"><span class="lab"><font color="Green" title="E114">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:121_1_5_1_1_2"/><span class="lab"><font color="Green" title="E116">A30</font></span>: 
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/><div class="add"><a NAME="E2:121_1_5_1_1_2"/><span class="kw">then </span>
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a2</font>, <font color="Maroon" title="c18">b2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:121_1_5_1_1_2"/><span class="lab"><font color="Green" title="E117">A31</font></span>: 
<font color="Maroon" title="c17">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">b2</font>
 <span class="kw">and </span><br/><a NAME="E5:121_1_5_1_1_2"/><span class="lab"><font color="Green" title="E118">A32</font></span>: 
 not <font color="Maroon" title="c17">a2</font>,<font color="Maroon" title="c18">b2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E6:121_1_5_1_1_2"/><span class="lab"><font color="Green" title="E119">A33</font></span>: 
<font color="Maroon" title="c17">a2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">and </span><br/><a NAME="E7:121_1_5_1_1_2"/><span class="lab"><font color="Green" title="E120">A34</font></span>: 
<font color="Maroon" title="c18">b2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E27:121_1"><span class="lab"><font color="Green" title="E113">A27</font></span></a>, <a class="txt" href="#E1:121_1_5_1_1_2"><span class="lab"><font color="Green" title="E116">A30</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:121_1_5_1_1_2_1"/><span class="lab"><font color="Green" title="E121">A35</font></span>: 
<font color="Maroon" title="c17">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">and </span><br/><a NAME="E2:121_1_5_1_1_2_1"/><span class="lab"><font color="Green" title="E122">A36</font></span>: 
<font color="Maroon" title="c18">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c19">a4</font> = <font color="Maroon" title="c17">a2</font>, <font color="Maroon" title="c20">b4</font> = <font color="Maroon" title="c18">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E121">A35</font></span></a>, <a class="txt" href="#E2:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E122">A36</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><a NAME="E4:121_1_5_1_1_2_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="txt" href="#E11:121_1"><span class="lab"><font color="Green" title="E101">A9</font></span></a>, <a class="txt" href="#E1:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E121">A35</font></span></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></span>;<br/><a NAME="E5:121_1_5_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A37</font></span>: 
<font color="Maroon" title="c19">a4</font>,<font color="Maroon" title="c20">b4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121_1_5_1_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">a3</font> = <font color="Maroon" title="c17">a2</font>, <font color="Maroon" title="c22">b3</font> = <font color="Maroon" title="c18">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E121">A35</font></span></a>, <a class="txt" href="#E2:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E122">A36</font></span></a></span>;<br/><a NAME="E7:121_1_5_1_1_2_1"/>
 not <font color="Maroon" title="c21">a3</font>,<font color="Maroon" title="c22">b3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:121_1"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="#E5:121_1_5_1_1_2"><span class="lab"><font color="Green" title="E118">A32</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><span class="kw">hence </span><a NAME="E8:121_1_5_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E121">A35</font></span></a>, <a class="txt" href="#E5:121_1_5_1_1_2_1"><span class="lab"><font color="Green" title="E123">A37</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:121_1_5_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> or <font color="Maroon" title="c17">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_1_5"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:121_1_5_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:121_1_5_1_1_2"><span class="lab"><font color="Green" title="E119">A33</font></span></a>, <a class="txt" href="#E7:121_1_5_1_1_2"><span class="lab"><font color="Green" title="E120">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">cc</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:121_1_5"/><span class="lab"><font color="Green" title="E116">A38</font></span>: 
<font color="Maroon" title="c17">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">and </span><br/><a NAME="E8:121_1_5"/><span class="lab"><font color="Green" title="E117">A39</font></span>: 
<font color="Maroon" title="c17">cc</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">cc</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:121_1"><span class="lab"><font color="Green" title="E108">A19</font></span></a>, <a class="txt" href="#E7:121_1_5"><span class="lab"><font color="Green" title="E116">A38</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_5_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Gk1</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="c18">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:121_1_5_2"/><span class="lab"><font color="Green" title="E118">A40</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Gk1</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="c18">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span><br/><span class="kw">assume </span><a NAME="E2:121_1_5_2"/><span class="lab"><font color="Green" title="E119">A41</font></span>: 
 not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:121_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A42</font></span>: 
 not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:121_1_5_2"/><span class="lab"><font color="Green" title="E121">A43</font></span>: 
 not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_5_2"><span class="lab"><font color="Green" title="E119">A41</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_5_2"><span class="lab"><font color="Green" title="E118">A40</font></span></a></span>;<br/><a NAME="E6:121_1_5_2"/>
<font color="Maroon" title="c18">c</font>,<font color="Maroon" title="c20">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_5_2"><span class="lab"><font color="Green" title="E118">A40</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:121_1_5_2"/><span class="lab"><font color="Green" title="E122">A44</font></span>: 
<font color="Maroon" title="c21">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">c</font>,<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c2">Gk1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E9:121_1_5_2"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:121_1"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E4:121_1_5_2"><span class="lab"><font color="Green" title="E121">A43</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E10:121_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">c</font>,<font color="Maroon" title="c20">x</font>, the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:121_1"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E7:121_1_5"><span class="lab"><font color="Green" title="E116">A38</font></span></a>, <a class="txt" href="#E8:121_1_5_2"><span class="lab"><font color="Green" title="E122">A44</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:121_1_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_5"><span class="lab"><font color="Green" title="E116">A38</font></span></a>, <a class="txt" href="#E3:121_1_5_2"><span class="lab"><font color="Green" title="E120">A42</font></span></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:121_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A45</font></span>: 
<font color="Maroon" title="c2">Gk1</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="c18">c</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> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:121_1_5"/>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_5"><span class="lab"><font color="Green" title="E116">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E13:121_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:121_1"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="#E8:121_1_5"><span class="lab"><font color="Green" title="E117">A39</font></span></a>, <a class="txt" href="#E11:121_1_5"><span class="lab"><font color="Green" title="E118">A45</font></span></a>, <a class="ref" href="chord.html#T67" target="_self" title="CHORD:th.67">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:121_1_5"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_5"><span class="lab"><font color="Green" title="E116">A38</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font><span class="kw"> such that </span><br/><a NAME="E30:121_1"/><span class="lab"><font color="Green" title="E114">A46</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">sb</font>
 <span class="kw">and </span><br/><a NAME="E31:121_1"/><span class="lab"><font color="Green" title="E115">A47</font></span>: 
<font color="Maroon" title="c15">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 ;<br/>
<a NAME="E32:121_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></span>;<br/>
<a NAME="E33:121_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A48</font></span>: 
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:121_1"><span class="lab"><font color="Green" title="E107">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E34:121_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c16">aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:121_1_6"/><span class="lab"><font color="Green" title="E117">A49</font></span>: 
<font color="Maroon" title="c16">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E3:121_1_6"/><span class="lab"><font color="Green" title="E118">A50</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:121_1"><span class="lab"><font color="Green" title="E104">A12</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c16">aa</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_6"><span class="lab"><font color="Green" title="E117">A49</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:121_1_6"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_6_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:121_1_6_1_1"/>
(  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  or  not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:121_1_6_1_1_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/><div class="add"><a NAME="E2:121_1_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T64" target="_self" title="CHORD:th.64">Th64</a></span>;<br/><span class="kw">hence </span><a NAME="E3:121_1_6_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_6"><span class="lab"><font color="Green" title="E117">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:121_1_6_1_1_2"/><span class="lab"><font color="Green" title="E119">A51</font></span>: 
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/><div class="add"><a NAME="E2:121_1_6_1_1_2"/><span class="kw">then </span>
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">a2</font>, <font color="Maroon" title="c19">b2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:121_1_6_1_1_2"/><span class="lab"><font color="Green" title="E120">A52</font></span>: 
<font color="Maroon" title="c18">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b2</font>
 <span class="kw">and </span><br/><a NAME="E5:121_1_6_1_1_2"/><span class="lab"><font color="Green" title="E121">A53</font></span>: 
 not <font color="Maroon" title="c18">a2</font>,<font color="Maroon" title="c19">b2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E6:121_1_6_1_1_2"/><span class="lab"><font color="Green" title="E122">A54</font></span>: 
<font color="Maroon" title="c18">a2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">and </span><br/><a NAME="E7:121_1_6_1_1_2"/><span class="lab"><font color="Green" title="E123">A55</font></span>: 
<font color="Maroon" title="c19">b2</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E33:121_1"><span class="lab"><font color="Green" title="E116">A48</font></span></a>, <a class="txt" href="#E1:121_1_6_1_1_2"><span class="lab"><font color="Green" title="E119">A51</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_6_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:121_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E124">A56</font></span>: 
<font color="Maroon" title="c18">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">and </span><br/><a NAME="E2:121_1_6_1_1_2_1"/><span class="lab"><font color="Green" title="E125">A57</font></span>: 
<font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c20">a4</font> = <font color="Maroon" title="c18">a2</font>, <font color="Maroon" title="c21">b4</font> = <font color="Maroon" title="c19">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E124">A56</font></span></a>, <a class="txt" href="#E2:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E125">A57</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><a NAME="E4:121_1_6_1_1_2_1"/>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121_1"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E9:121_1"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="txt" href="#E11:121_1"><span class="lab"><font color="Green" title="E101">A9</font></span></a>, <a class="txt" href="#E1:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E124">A56</font></span></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></span>;<br/><a NAME="E5:121_1_6_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A58</font></span>: 
<font color="Maroon" title="c20">a4</font>,<font color="Maroon" title="c21">b4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121_1_6_1_1_2"><span class="lab"><font color="Green" title="E120">A52</font></span></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">a3</font> = <font color="Maroon" title="c18">a2</font>, <font color="Maroon" title="c23">b3</font> = <font color="Maroon" title="c19">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E124">A56</font></span></a>, <a class="txt" href="#E2:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E125">A57</font></span></a></span>;<br/><a NAME="E7:121_1_6_1_1_2_1"/>
 not <font color="Maroon" title="c22">a3</font>,<font color="Maroon" title="c23">b3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:121_1"><span class="lab"><font color="Green" title="E104">A12</font></span></a>, <a class="txt" href="#E5:121_1_6_1_1_2"><span class="lab"><font color="Green" title="E121">A53</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><span class="kw">hence </span><a NAME="E8:121_1_6_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E124">A56</font></span></a>, <a class="txt" href="#E5:121_1_6_1_1_2_1"><span class="lab"><font color="Green" title="E126">A58</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:121_1_6_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> or <font color="Maroon" title="c18">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_1_6"><span class="lab"><font color="Green" title="E118">A50</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:121_1_6_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:121_1_6_1_1_2"><span class="lab"><font color="Green" title="E122">A54</font></span></a>, <a class="txt" href="#E7:121_1_6_1_1_2"><span class="lab"><font color="Green" title="E123">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">cc</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c3">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:121_1_6"/><span class="lab"><font color="Green" title="E119">A59</font></span>: 
<font color="Maroon" title="c18">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">and </span><br/><a NAME="E8:121_1_6"/><span class="lab"><font color="Green" title="E120">A60</font></span>: 
<font color="Maroon" title="c18">cc</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c18">cc</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:121_1"><span class="lab"><font color="Green" title="E103">A11</font></span></a>, <a class="txt" href="#E7:121_1_6"><span class="lab"><font color="Green" title="E119">A59</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_6_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Gk1</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="c19">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:121_1_6_2"/><span class="lab"><font color="Green" title="E121">A61</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Gk1</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="c19">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span><br/><span class="kw">assume </span><a NAME="E2:121_1_6_2"/><span class="lab"><font color="Green" title="E122">A62</font></span>: 
 not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:121_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A63</font></span>: 
 not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:121_1_6_2"/><span class="lab"><font color="Green" title="E124">A64</font></span>: 
 not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121_1_6_2"><span class="lab"><font color="Green" title="E122">A62</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c20">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_6_2"><span class="lab"><font color="Green" title="E121">A61</font></span></a></span>;<br/><a NAME="E6:121_1_6_2"/>
<font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c21">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_6_2"><span class="lab"><font color="Green" title="E121">A61</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:121_1_6_2"/><span class="lab"><font color="Green" title="E125">A65</font></span>: 
<font color="Maroon" title="c22">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c2">Gk1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E9:121_1_6_2"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:121_1"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E4:121_1_6_2"><span class="lab"><font color="Green" title="E124">A64</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E10:121_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c19">c</font>,<font color="Maroon" title="c21">x</font>, the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:121_1"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E7:121_1_6"><span class="lab"><font color="Green" title="E119">A59</font></span></a>, <a class="txt" href="#E8:121_1_6_2"><span class="lab"><font color="Green" title="E125">A65</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:121_1_6_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_6"><span class="lab"><font color="Green" title="E119">A59</font></span></a>, <a class="txt" href="#E3:121_1_6_2"><span class="lab"><font color="Green" title="E123">A63</font></span></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:121_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A66</font></span>: 
<font color="Maroon" title="c2">Gk1</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="c19">c</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> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:121_1_6"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_6"><span class="lab"><font color="Green" title="E119">A59</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E13:121_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c19">c</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:121_1"><span class="lab"><font color="Green" title="E104">A12</font></span></a>, <a class="txt" href="#E8:121_1_6"><span class="lab"><font color="Green" title="E120">A60</font></span></a>, <a class="txt" href="#E11:121_1_6"><span class="lab"><font color="Green" title="E121">A66</font></span></a>, <a class="ref" href="chord.html#T67" target="_self" title="CHORD:th.67">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:121_1_6"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:121_1_6"><span class="lab"><font color="Green" title="E119">A59</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font><span class="kw"> such that </span><br/><a NAME="E36:121_1"/><span class="lab"><font color="Green" title="E117">A67</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">and </span><br/><a NAME="E37:121_1"/><span class="lab"><font color="Green" title="E118">A68</font></span>: 
<font color="Maroon" title="c16">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 ;<br/>
<div><span class="lab"><font color="Green" title="E119">A69</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">aa</font> = <font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c18">bb</font> = <font color="Maroon" title="c15">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:121_1"><span class="lab"><font color="Green" title="E117">A67</font></span></a>, <a class="txt" href="#E30:121_1"><span class="lab"><font color="Green" title="E114">A46</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:121_1_7"/>
<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c15">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:121_1_7"/><span class="lab"><font color="Green" title="E120">A70</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c2">Gk1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E5:121_1_7"/>
<font color="Maroon" title="c19">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">aa</font>,<font color="Maroon" title="c18">bb</font>, the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:121_1"><span class="lab"><font color="Green" title="E112">A26</font></span></a>, <a class="txt" href="#E4:121_1_7"><span class="lab"><font color="Green" title="E120">A70</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><a NAME="E6:121_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c6">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:121_1"><span class="lab"><font color="Green" title="E117">A67</font></span></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></span>;<br/><span class="kw">hence </span><a NAME="E7:121_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:121_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="txt" href="#E30:121_1"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E39:121_1"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:121_1"><span class="lab"><font color="Green" title="E105">A13</font></span></a>, <a class="txt" href="#E36:121_1"><span class="lab"><font color="Green" title="E117">A67</font></span></a>, <a class="txt" href="#E30:121_1"><span class="lab"><font color="Green" title="E114">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:121_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:121_1"><span class="lab"><font color="Green" title="E118">A68</font></span></a>, <a class="txt" href="#E31:121_1"><span class="lab"><font color="Green" title="E115">A47</font></span></a>, <a class="txt" href="#E38:121_1"><span class="lab"><font color="Green" title="E119">A69</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:121"/><span class="lab"><font color="Green" title="E94">A71</font></span>: 
 for <font color="Olive" title="b1">G</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="chord.html#S1" target="_self" title="CHORD:sch.1">CHORD:sch 1</a>(<a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E93">A1</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>   <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:121"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E4:121"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121"><span class="lab"><font color="Green" title="E94">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
