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<div class="add">

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E1:122"/><i><font color="Green" title="E94">A1</font></i>: 
not <font color="Maroon" title="c1">G</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/>

<b>defpred </b><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>] <b>means </b>( 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="E2:122"/><i><font color="Green" title="E95">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">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#K16" title="GLIB_000:func.16">.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#K16" title="GLIB_000:func.16">.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="122_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:122_1"/><i><font color="Green" title="E96">A3</font></i>: 
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#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c3">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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:122_1"/><i><font color="Green" title="E97">A4</font></i>: 
<font color="Maroon" title="c3">Gk1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <b>and </b><br/><a NAME="E3:122_1"/><i><font color="Green" title="E98">A5</font></i>: 
not <font color="Maroon" title="c3">Gk1</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 <b>and </b><br/><a NAME="E4:122_1"/><i><font color="Green" title="E99">A6</font></i>: 
<font color="Maroon" title="c3">Gk1</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>
 <b>and </b><br/><a NAME="E5:122_1"/><i><font color="Green" title="E100">A7</font></i>: 
not <font color="Maroon" title="c3">Gk1</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">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> <b>by </b><i><a class="txt" href="chord.html#E3:122_1"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="chord.html#E4:122_1"><i><font color="Green" title="E99">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G</font><b> such that </b><br/><a NAME="E8:122_1"/><i><font color="Green" title="E101">A8</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E9:122_1"/><i><font color="Green" title="E102">A9</font></i>: 
not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a>, <a class="txt" href="chord.html#E5:122_1"><i><font color="Green" title="E100">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font><b> such that </b><br/><a NAME="E11:122_1"/><i><font color="Green" title="E103">A10</font></i>: 
<font color="Maroon" title="c7">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>
 <b>by </b><i><a class="ref" href="chord.html#T79" target="_self" title="CHORD:th.79">Th79</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">Gs</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">Gns</font> being   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">S</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">sa</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c11">sb</font> = <font color="Maroon" title="c6">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gns</font> <b>by </b><i><a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="chord.html#T77" target="_self" title="CHORD:th.77">Th77</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>;<br/>
<b>set </b><font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font>;<br/>
<b>consider </b><font color="Maroon" title="c14">Gas</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c15">Gbs</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font></span>)</span>;<br/>
<a NAME="E17:122_1"/>
( <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">Gns</font> &amp; <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">Gns</font> )
 
;<br/>
<a NAME="E18:122_1"/><b>then </b><i><font color="Green" title="E104">A11</font></i>: 
( <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">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="c4">G</font> &amp; <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">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="c4">G</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:122_1"/><b>then </b><i><font color="Green" title="E105">A12</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">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="c4">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">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="c4">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E20:122_1"/><i><font color="Green" title="E106">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">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> 
 
<b>by </b><i><a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="chord.html#T76" target="_self" title="CHORD:th.76">Th76</a></i>;<br/>
<div><i><font color="Green" title="E107">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1"/><i><font color="Green" title="E108">A15</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 ;<br/><a NAME="E2:122_1_1"/>
not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/><a NAME="E3:122_1_1"/><b>then </b><i><font color="Green" title="E109">A16</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:122_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>;<br/><b>hence </b><a NAME="E5:122_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E20:122_1"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="chord.html#E3:122_1_1"><i><font color="Green" title="E109">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E108">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_2"/><i><font color="Green" title="E109">A18</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 ;<br/><a NAME="E2:122_1_2"/>
not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/><a NAME="E3:122_1_2"/><b>then </b><i><font color="Green" title="E110">A19</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_2"><i><font color="Green" title="E109">A18</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:122_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>;<br/><b>hence </b><a NAME="E5:122_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E20:122_1"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="chord.html#E3:122_1_2"><i><font color="Green" title="E110">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:122_1"/>
not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/>
<a NAME="E24:122_1"/><b>then </b>
<font color="Maroon" title="c5">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="c4">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E25:122_1"/><b>then </b><i><font color="Green" title="E109">A20</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">Gns</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="c4">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E26:122_1"/><i><font color="Green" title="E110">A21</font></i>: 
<font color="Maroon" title="c14">Gas</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E2:122_1"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="glib_000.html#T78" title="GLIB_000:th.78">GLIB_000:78</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_3"/>
<font color="Maroon" title="c14">Gas</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 ;<br/><a NAME="E2:122_1_3"/><b>then </b><i><font color="Green" title="E111">A22</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">Gas</font> <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="c4">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/><a NAME="E3:122_1_3"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">Gas</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E4:122_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E21:122_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="chord.html#E2:122_1_3"><i><font color="Green" title="E111">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:122_1"/><b>then </b><i><font color="Green" title="E111">A23</font></i>: 
<font color="Maroon" title="c14">Gas</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E26:122_1"><i><font color="Green" title="E110">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E29:122_1"/><i><font color="Green" title="E112">A24</font></i>: 
<font color="Maroon" title="c15">Gbs</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E2:122_1"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="glib_000.html#T78" title="GLIB_000:th.78">GLIB_000:78</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_4"/>
<font color="Maroon" title="c15">Gbs</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 ;<br/><a NAME="E2:122_1_4"/><b>then </b><i><font color="Green" title="E113">A25</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">Gbs</font> <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="c4">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/><a NAME="E3:122_1_4"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">Gbs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E4:122_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E22:122_1"><i><font color="Green" title="E108">A17</font></i></a>, <a class="txt" href="chord.html#E2:122_1_4"><i><font color="Green" title="E113">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:122_1"/><b>then </b><i><font color="Green" title="E113">A26</font></i>: 
<font color="Maroon" title="c15">Gbs</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E29:122_1"><i><font color="Green" title="E112">A24</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E32:122_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="c3">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">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="122_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c16">aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:122_1_5"/><i><font color="Green" title="E114">A27</font></i>: 
<font color="Maroon" title="c16">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:122_1_5"/><i><font color="Green" title="E115">A28</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">Gas</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>then reconsider </b><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 <font color="Maroon" title="c14">Gas</font> <b>by </b><i><a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green" title="E114">A27</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:122_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 <font color="Maroon" title="c14">Gas</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">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="122_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_5_1_1"/>
( <font color="Maroon" title="c14">Gas</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a> or  not <font color="Maroon" title="c14">Gas</font> 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="122_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:122_1_5_1_1_1"/>
<font color="Maroon" title="c14">Gas</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">a</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="ref" href="chord.html#T64" target="_self" title="CHORD:th.64">Th64</a></i>;<br/><b>hence </b><a NAME="E3:122_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 <font color="Maroon" title="c14">Gas</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> &amp; <font color="Olive" title="b1">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1_5"><i><font color="Green" title="E114">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:122_1_5_1_1_2"/><i><font color="Green" title="E116">A29</font></i>: 
not <font color="Maroon" title="c14">Gas</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_5_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">Gas</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 <b>by </b><i><a class="ref" href="chord.html#T61" target="_self" title="CHORD:th.61">Th61</a></i>;<br/><b>then consider </b><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 <font color="Maroon" title="c14">Gas</font><b> such that </b><br/><a NAME="E4:122_1_5_1_1_2"/><i><font color="Green" title="E117">A30</font></i>: 
<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>
 <b>and </b><br/><a NAME="E5:122_1_5_1_1_2"/><i><font color="Green" title="E118">A31</font></i>: 
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> 
 <b>and </b><br/><a NAME="E6:122_1_5_1_1_2"/><i><font color="Green" title="E119">A32</font></i>: 
( <font color="Maroon" title="c18">a2</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> &amp; <font color="Maroon" title="c19">b2</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_1_1_2"><i><font color="Green" title="E116">A29</font></i></a>, <a class="txt" href="chord.html#E28:122_1"><i><font color="Green" title="E111">A23</font></i></a>, <a class="txt" href="chord.html#E1:122_1"><i><font color="Green" title="E96">A3</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_5_1_1_2_1"/><i><font color="Green" title="E120">A33</font></i>: 
( <font color="Maroon" title="c18">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c20">a3</font> = <font color="Maroon" title="c18">a2</font>, <font color="Maroon" title="c21">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="c4">G</font> ;<br/><a NAME="E3:122_1_5_1_1_2_1"/><i><font color="Green" title="E121">A34</font></i>: 
not <font color="Maroon" title="c20">a3</font>,<font color="Maroon" title="c21">b3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:122_1_5_1_1_2"><i><font color="Green" title="E118">A31</font></i></a>, <a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/><a NAME="E4:122_1_5_1_1_2_1"/><i><font color="Green" title="E122">A35</font></i>: 
<font color="Maroon" title="c8">Gs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_1_1_2_1"><i><font color="Green" title="E120">A33</font></i></a>, <a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="chord.html#E11:122_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">a4</font> = <font color="Maroon" title="c18">a2</font>, <font color="Maroon" title="c23">b4</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="c8">Gs</font> <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_1_1_2_1"><i><font color="Green" title="E120">A33</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:122_1_5_1_1_2_1"/>
<font color="Maroon" title="c22">a4</font>,<font color="Maroon" title="c23">b4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:122_1_5_1_1_2_1"><i><font color="Green" title="E122">A35</font></i></a>, <a class="txt" href="chord.html#E4:122_1_5_1_1_2"><i><font color="Green" title="E117">A30</font></i></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E7:122_1_5_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_5_1_1_2_1"><i><font color="Green" title="E121">A34</font></i></a>, <a class="txt" href="chord.html#E1:122_1_5_1_1_2_1"><i><font color="Green" title="E120">A33</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/></div><b>end;</b></div><a NAME="E8:122_1_5_1_1_2"/><b>then </b>
( <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> or <font color="Maroon" title="c18">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> )
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_5"><i><font color="Green" title="E115">A28</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:122_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 <font color="Maroon" title="c14">Gas</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> &amp; <font color="Olive" title="b1">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E6:122_1_5_1_1_2"><i><font color="Green" title="E119">A32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c18">cc</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c14">Gas</font><b> such that </b><br/><a NAME="E7:122_1_5"/><i><font color="Green" title="E116">A36</font></i>: 
<font color="Maroon" title="c18">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>
 <b>and </b><br/><a NAME="E8:122_1_5"/><i><font color="Green" title="E117">A37</font></i>: 
<font color="Maroon" title="c18">cc</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 ;<br/>
<b>reconsider </b><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="c3">Gk1</font> <b>by </b><i><a class="txt" href="chord.html#E18:122_1"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="chord.html#E7:122_1_5"><i><font color="Green" title="E116">A36</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:122_1_5_2"/><i><font color="Green" title="E118">A38</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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>
 ;<br/><b>assume </b><a NAME="E2:122_1_5_2"/>
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"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 ;<br/><a NAME="E3:122_1_5_2"/><b>then </b><i><font color="Green" title="E119">A39</font></i>: 
( not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> &amp; not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><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="c3">Gk1</font> <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_2"><i><font color="Green" title="E118">A38</font></i></a></i>;<br/><a NAME="E5:122_1_5_2"/><i><font color="Green" title="E120">A40</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E25:122_1"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E3:122_1_5_2"><i><font color="Green" title="E119">A39</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:122_1_5_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> 
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_5_2"><i><font color="Green" title="E118">A38</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:122_1_5_2"/><i><font color="Green" title="E121">A41</font></i>: 
<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="c3">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E9:122_1_5_2"/>
<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="c9">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E25:122_1"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E5:122_1_5_2"><i><font color="Green" title="E120">A40</font></i></a>, <a class="txt" href="chord.html#E8:122_1_5_2"><i><font color="Green" title="E121">A41</font></i></a>, <a class="txt" href="chord.html#E7:122_1_5"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E10:122_1_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_5_2"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="chord.html#E7:122_1_5"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:122_1_5"/><b>then </b><i><font color="Green" title="E118">A42</font></i>: 
<font color="Maroon" title="c3">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"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:122_1_5"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E7:122_1_5"><i><font color="Green" title="E116">A36</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:122_1_5"/><b>then </b>
<font color="Maroon" title="c19">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 
<b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="txt" href="chord.html#E8:122_1_5"><i><font color="Green" title="E117">A37</font></i></a>, <a class="txt" href="chord.html#E11:122_1_5"><i><font color="Green" title="E118">A42</font></i></a>, <a class="ref" href="chord.html#T67" target="_self" title="CHORD:th.67">Th67</a></i>;<br/>
<b>hence </b><a NAME="E14:122_1_5"/>
 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="c3">Gk1</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> &amp; <font color="Olive" title="b1">a</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E7:122_1_5"><i><font color="Green" title="E116">A36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><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="c3">Gk1</font><b> such that </b><br/><a NAME="E34:122_1"/><i><font color="Green" title="E114">A43</font></i>: 
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font> &amp; <font color="Maroon" title="c16">a</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 ;<br/>
<a NAME="E35:122_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="c3">Gk1</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">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="122_1_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c17">aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:122_1_6"/><i><font color="Green" title="E115">A44</font></i>: 
<font color="Maroon" title="c17">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:122_1_6"/><i><font color="Green" title="E116">A45</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">Gbs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c17">aa</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c15">Gbs</font> <b>by </b><i><a class="txt" href="chord.html#E2:122_1_6"><i><font color="Green" title="E115">A44</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:122_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 <font color="Maroon" title="c15">Gbs</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">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="122_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_6_1_1"/>
( <font color="Maroon" title="c15">Gbs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a> or  not <font color="Maroon" title="c15">Gbs</font> 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="122_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:122_1_6_1_1_1"/>
<font color="Maroon" title="c15">Gbs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">a</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="ref" href="chord.html#T64" target="_self" title="CHORD:th.64">Th64</a></i>;<br/><b>hence </b><a NAME="E3:122_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 <font color="Maroon" title="c15">Gbs</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> &amp; <font color="Olive" title="b1">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E2:122_1_6"><i><font color="Green" title="E115">A44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:122_1_6_1_1_2"/><i><font color="Green" title="E117">A46</font></i>: 
not <font color="Maroon" title="c15">Gbs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 ;<br/><div class="add"><a NAME="E2:122_1_6_1_1_2"/><b>then </b>
not <font color="Maroon" title="c15">Gbs</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 <b>by </b><i><a class="ref" href="chord.html#T61" target="_self" title="CHORD:th.61">Th61</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">a2</font>, <font color="Maroon" title="c20">b2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c15">Gbs</font><b> such that </b><br/><a NAME="E4:122_1_6_1_1_2"/><i><font color="Green" title="E118">A47</font></i>: 
<font color="Maroon" title="c19">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">b2</font>
 <b>and </b><br/><a NAME="E5:122_1_6_1_1_2"/><i><font color="Green" title="E119">A48</font></i>: 
not <font color="Maroon" title="c19">a2</font>,<font color="Maroon" title="c20">b2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>and </b><br/><a NAME="E6:122_1_6_1_1_2"/><i><font color="Green" title="E120">A49</font></i>: 
( <font color="Maroon" title="c19">a2</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> &amp; <font color="Maroon" title="c20">b2</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_6_1_1_2"><i><font color="Green" title="E117">A46</font></i></a>, <a class="txt" href="chord.html#E31:122_1"><i><font color="Green" title="E113">A26</font></i></a>, <a class="txt" href="chord.html#E1:122_1"><i><font color="Green" title="E96">A3</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_6_1_1_2_1"/><i><font color="Green" title="E121">A50</font></i>: 
( <font color="Maroon" title="c19">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">a3</font> = <font color="Maroon" title="c19">a2</font>, <font color="Maroon" title="c22">b3</font> = <font color="Maroon" title="c20">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G</font> ;<br/><a NAME="E3:122_1_6_1_1_2_1"/><i><font color="Green" title="E122">A51</font></i>: 
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> 
 <b>by </b><i><a class="txt" href="chord.html#E5:122_1_6_1_1_2"><i><font color="Green" title="E119">A48</font></i></a>, <a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/><a NAME="E4:122_1_6_1_1_2_1"/><i><font color="Green" title="E123">A52</font></i>: 
<font color="Maroon" title="c8">Gs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_6_1_1_2_1"><i><font color="Green" title="E121">A50</font></i></a>, <a class="txt" href="chord.html#E8:122_1"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="chord.html#E9:122_1"><i><font color="Green" title="E102">A9</font></i></a>, <a class="txt" href="chord.html#E11:122_1"><i><font color="Green" title="E103">A10</font></i></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">a4</font> = <font color="Maroon" title="c19">a2</font>, <font color="Maroon" title="c24">b4</font> = <font color="Maroon" title="c20">b2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c8">Gs</font> <b>by </b><i><a class="txt" href="chord.html#E1:122_1_6_1_1_2_1"><i><font color="Green" title="E121">A50</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:122_1_6_1_1_2_1"/>
<font color="Maroon" title="c23">a4</font>,<font color="Maroon" title="c24">b4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:122_1_6_1_1_2_1"><i><font color="Green" title="E123">A52</font></i></a>, <a class="txt" href="chord.html#E4:122_1_6_1_1_2"><i><font color="Green" title="E118">A47</font></i></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E7:122_1_6_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_6_1_1_2_1"><i><font color="Green" title="E122">A51</font></i></a>, <a class="txt" href="chord.html#E1:122_1_6_1_1_2_1"><i><font color="Green" title="E121">A50</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/></div><b>end;</b></div><a NAME="E8:122_1_6_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> or <font color="Maroon" title="c19">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> )
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_6"><i><font color="Green" title="E116">A45</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:122_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 <font color="Maroon" title="c15">Gbs</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> &amp; <font color="Olive" title="b1">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E6:122_1_6_1_1_2"><i><font color="Green" title="E120">A49</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c19">cc</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c15">Gbs</font><b> such that </b><br/><a NAME="E7:122_1_6"/><i><font color="Green" title="E117">A53</font></i>: 
<font color="Maroon" title="c19">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font>
 <b>and </b><br/><a NAME="E8:122_1_6"/><i><font color="Green" title="E118">A54</font></i>: 
<font color="Maroon" title="c19">cc</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">c</font> = <font color="Maroon" title="c19">cc</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">Gk1</font> <b>by </b><i><a class="txt" href="chord.html#E18:122_1"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="chord.html#E7:122_1_6"><i><font color="Green" title="E117">A53</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:122_1_6_2"/><i><font color="Green" title="E119">A55</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c20">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>assume </b><a NAME="E2:122_1_6_2"/>
not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 ;<br/><a NAME="E3:122_1_6_2"/><b>then </b><i><font color="Green" title="E120">A56</font></i>: 
( not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> &amp; not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">Gk1</font> <b>by </b><i><a class="txt" href="chord.html#E1:122_1_6_2"><i><font color="Green" title="E119">A55</font></i></a></i>;<br/><a NAME="E5:122_1_6_2"/><i><font color="Green" title="E121">A57</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c9">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E25:122_1"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E3:122_1_6_2"><i><font color="Green" title="E120">A56</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:122_1_6_2"/>
<font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c22">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:122_1_6_2"><i><font color="Green" title="E119">A55</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:122_1_6_2"/><i><font color="Green" title="E122">A58</font></i>: 
<font color="Maroon" title="c23">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c3">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E9:122_1_6_2"/>
<font color="Maroon" title="c23">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c20">c</font>,<font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c9">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:122_1_6"><i><font color="Green" title="E117">A53</font></i></a>, <a class="txt" href="chord.html#E25:122_1"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E5:122_1_6_2"><i><font color="Green" title="E121">A57</font></i></a>, <a class="txt" href="chord.html#E8:122_1_6_2"><i><font color="Green" title="E122">A58</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E10:122_1_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:122_1_6_2"><i><font color="Green" title="E120">A56</font></i></a>, <a class="txt" href="chord.html#E7:122_1_6"><i><font color="Green" title="E117">A53</font></i></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:122_1_6"/><b>then </b><i><font color="Green" title="E119">A59</font></i>: 
<font color="Maroon" title="c3">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="c20">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"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:122_1_6"/>
<font color="Maroon" title="c20">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E7:122_1_6"><i><font color="Green" title="E117">A53</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:122_1_6"/><b>then </b>
<font color="Maroon" title="c20">c</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 
<b>by </b><i><a class="txt" href="chord.html#E19:122_1"><i><font color="Green" title="E105">A12</font></i></a>, <a class="txt" href="chord.html#E8:122_1_6"><i><font color="Green" title="E118">A54</font></i></a>, <a class="txt" href="chord.html#E11:122_1_6"><i><font color="Green" title="E119">A59</font></i></a>, <a class="ref" href="chord.html#T67" target="_self" title="CHORD:th.67">Th67</a></i>;<br/>
<b>hence </b><a NAME="E14:122_1_6"/>
 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="c3">Gk1</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> &amp; <font color="Olive" title="b1">b</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 <b>by </b><i><a class="txt" href="chord.html#E7:122_1_6"><i><font color="Green" title="E117">A53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c17">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">Gk1</font><b> such that </b><br/><a NAME="E37:122_1"/><i><font color="Green" title="E115">A60</font></i>: 
( <font color="Maroon" title="c17">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c11">sb</font> &amp; <font color="Maroon" title="c17">b</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 ;<br/>
<div><i><font color="Green" title="E116">A61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_7"/>
<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c18">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:122_1_7"/><i><font color="Green" title="E117">A62</font></i>: 
<font color="Maroon" title="c18">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="c17">b</font>,<font color="Maroon" title="c3">Gk1</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">aa</font> = <font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c20">bb</font> = <font color="Maroon" title="c17">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gns</font> <b>by </b><i><a class="txt" href="chord.html#E34:122_1"><i><font color="Green" title="E114">A43</font></i></a>, <a class="txt" href="chord.html#E37:122_1"><i><font color="Green" title="E115">A60</font></i></a></i>;<br/><a NAME="E5:122_1_7"/>
<font color="Maroon" title="c18">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c19">aa</font>,<font color="Maroon" title="c20">bb</font>,<font color="Maroon" title="c9">Gns</font>
 <b>by </b><i><a class="txt" href="chord.html#E25:122_1"><i><font color="Green" title="E109">A20</font></i></a>, <a class="txt" href="chord.html#E3:122_1_7"><i><font color="Green" title="E117">A62</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></i>;<br/><a NAME="E6:122_1_7"/><b>then </b>
<font color="Maroon" title="c20">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gns</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c10">sa</font>
 <b>by </b><i><a class="txt" href="chord.html#E34:122_1"><i><font color="Green" title="E114">A43</font></i></a>, <a class="ref" href="glib_002.html#T10" title="GLIB_002:th.10">GLIB_002:10</a></i>;<br/><b>hence </b><a NAME="E7:122_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E20:122_1"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="chord.html#E37:122_1"><i><font color="Green" title="E115">A60</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E39:122_1"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">b</font>
 
<b>by </b><i><a class="txt" href="chord.html#E20:122_1"><i><font color="Green" title="E106">A13</font></i></a>, <a class="txt" href="chord.html#E34:122_1"><i><font color="Green" title="E114">A43</font></i></a>, <a class="txt" href="chord.html#E37:122_1"><i><font color="Green" title="E115">A60</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E40:122_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="c3">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> )
 <b>by </b><i><a class="txt" href="chord.html#E34:122_1"><i><font color="Green" title="E114">A43</font></i></a>, <a class="txt" href="chord.html#E37:122_1"><i><font color="Green" title="E115">A60</font></i></a>, <a class="txt" href="chord.html#E38:122_1"><i><font color="Green" title="E116">A61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:122"/>
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>]
 
<b>from </b><i><a class="ref" href="chord.html#S1" target="_self" title="CHORD:sch.1">CHORD:sch 1</a>(<a class="txt" href="chord.html#E2:122"><i><font color="Green" title="E95">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:122"/>
 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> )
 <b>by </b><i><a class="txt" href="chord.html#E1:122"><i><font color="Green" title="E94">A1</font></i></a></i>;<br/>


</div>
