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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:96"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:96"/><i><font color="Green" title="E80">A2</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<b>assume </b><a NAME="E3:96"/><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">H</font> be   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c6">aa</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font>;<br/>



<b>assume </b><a NAME="E4:96"/><i><font color="Green" title="E82">A4</font></i>: 
<font color="Maroon" title="c6">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E5:96"/><i><font color="Green" title="E83">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">bb</font> = <font color="Maroon" title="c3">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font> <b>by </b><i><a class="txt" href="chord.html#E1:96"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="chord.html#E2:96"><i><font color="Green" title="E80">A2</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="c9">T</font> = <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">aa</font>;<br/>
<b>set </b><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">bb</font>;<br/>
<b>assume </b><a NAME="E7:96"/><i><font color="Green" title="E84">A6</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">aa</font> or  not <font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 ;<br/>

<a NAME="E8:96"/><i><font color="Green" title="E85">A7</font></i>: 
for <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/> ex <font color="Olive" title="b2">y</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="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:96_1"/><i><font color="Green" title="E86">A8</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:96_1"/><i><font color="Green" title="E87">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">k</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:96_1"/><i><font color="Green" title="E88">A10</font></i>: 
( <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font> &amp; <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:96_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/>
<a NAME="E5:96_1"/><i><font color="Green" title="E89">A11</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1_1"/>
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:96_1_1"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ;<br/><a NAME="E3:96_1_1"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:96_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E4:96_1_1"/><b>then </b>
not <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:96"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="chord.html#E2:96"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E5:96_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:96"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="chord.html#E2:96_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:96_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="chord.html#E5:96_1"><i><font color="Green" title="E89">A11</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="E8:96_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/>
<a NAME="E9:96_1"/><b>then </b><i><font color="Green" title="E90">A12</font></i>: 
3 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:96_1"/><b>then </b><i><font color="Green" title="E91">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E12:96_1"/>
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="txt" href="chord.html#E4:96_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="glib_001.html#T116" title="GLIB_001:th.116">GLIB_001:116</a></i>;<br/>
<a NAME="E13:96_1"/><b>then </b><i><font color="Green" title="E92">A14</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="txt" href="chord.html#E4:96_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:96_1"/><b>then </b><i><font color="Green" title="E93">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_2"/>
( <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  not <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2_1"><b>suppose </b></a><a NAME="E1:96_1_2_1"/><i><font color="Green" title="E94">A16</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:96_1_2_1"/>
( <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E10:96_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="chord.html#E14:96_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E3:96_1_2_1"/>
 ex <font color="Olive" title="b1">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E1:96_1_2_1"><i><font color="Green" title="E94">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2_2"><b>suppose </b></a><a NAME="E1:96_1_2_2"/>
not <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:96_1_2_2"/><i><font color="Green" title="E94">A17</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 ;<br/><b>set </b><font color="Maroon" title="c15">P</font> = <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font>;<br/><a NAME="E3:96_1_2_2"/><i><font color="Green" title="E95">A18</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font> )
 <b>by </b><i><a class="txt" href="chord.html#E14:96_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="chord.html#E9:96_1"><i><font color="Green" title="E90">A12</font></i></a></i>;<br/><a NAME="E4:96_1_2_2"/><b>then </b><i><font color="Green" title="E96">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E10:96_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E5:96_1_2_2"/><i><font color="Green" title="E97">A20</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> holds <br/>( not <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:96_1_2_2_1"/><i><font color="Green" title="E98">A21</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font>
 ;<br/>

<a NAME="E2:96_1_2_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E3:96_1_2_2_1"/><b>then </b>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:96_1_2_2_1"/><b>then </b><i><font color="Green" title="E99">A22</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">nu1</font> = <font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E6:96_1_2_2_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="chord.html#E1:96_1_2_2_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:96_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E8:96_1_2_2_1"/><b>then </b><i><font color="Green" title="E100">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">nu1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">nu1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2_1"><i><font color="Green" title="E99">A22</font></i></a>, <a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="txt" href="chord.html#E3:96_1_2_2"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1_2_2_1_1"/><i><font color="Green" title="E101">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/><a NAME="E2:96_1_2_2_1_1"/><i><font color="Green" title="E102">A25</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:96_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E103">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="txt" href="chord.html#E1:96_1_2_2_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:96_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E10:96_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="chord.html#E3:96_1_2_2"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="glib_001.html#T95" title="GLIB_001:th.95">GLIB_001:95</a></i>;<br/><a NAME="E5:96_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E104">A27</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E3:96_1_2_2_1_1"><i><font color="Green" title="E103">A26</font></i></a>, <a class="txt" href="chord.html#E2:96_1_2_2"><i><font color="Green" title="E94">A17</font></i></a></i>;<br/><a NAME="E6:96_1_2_2_1_1"/><i><font color="Green" title="E105">A28</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:96"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:96_1_2_2_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/><a NAME="E8:96_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E6:96_1_2_2_1_1"><i><font color="Green" title="E105">A28</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E9:96_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E106">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E1:96_1_2_2_1_1"><i><font color="Green" title="E101">A24</font></i></a>, <a class="txt" href="chord.html#E5:96_1_2_2_1_1"><i><font color="Green" title="E104">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="E10:96_1_2_2_1_1"/><i><font color="Green" title="E107">A30</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:96_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="chord.html#E1:96_1_2_2_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:96_1_2_2_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E1:96_1_2_2_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:96_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:96_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E9:96_1_2_2_1_1"><i><font color="Green" title="E106">A29</font></i></a>, <a class="txt" href="chord.html#E8:96_1_2_2_1"><i><font color="Green" title="E100">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E14:96_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:96_1_2_2_1_1"><i><font color="Green" title="E102">A25</font></i></a>, <a class="txt" href="chord.html#E10:96_1_2_2_1_1"><i><font color="Green" title="E107">A30</font></i></a>, <a class="ref" href="glib_001.html#T116" title="GLIB_001:th.116">GLIB_001:116</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:96_1_2_2_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E8:96_1_2_2_1"><i><font color="Green" title="E100">A23</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:96_1_2_2"/><b>then </b>
for <font color="Olive" title="b1">n</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> holds <br/> not <span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">HP</font> = <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">H</font> <b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="chord.html#T21" target="_self" title="CHORD:th.21">Th21</a></i>;<br/><a NAME="E8:96_1_2_2"/>
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:96_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E9:96_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:96_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="txt" href="chord.html#E5:96_1_2_2"><i><font color="Green" title="E97">A20</font></i></a></i>;<br/><a NAME="E10:96_1_2_2"/><b>then </b>
<font color="Maroon" title="c6">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">HP</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:96"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="chord.html#E9:96_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E11:96_1_2_2"/><b>then </b><i><font color="Green" title="E98">A31</font></i>: 
<font color="Maroon" title="c16">HP</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">aa</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T13" title="GLIB_002:th.13">GLIB_002:13</a></i>;<br/><a NAME="E12:96_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">HP</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E13:96_1_2_2"/><b>then </b><i><font color="Green" title="E99">A32</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">HP</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:96_1_2_2"><i><font color="Green" title="E97">A20</font></i></a></i>;<br/><a NAME="E14:96_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">Wj</font> = <font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E5:96_1_2_2"><i><font color="Green" title="E97">A20</font></i></a></i>;<br/><a NAME="E16:96_1_2_2"/>
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E13:96_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:96_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">W</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E18:96_1_2_2"/><b>then </b>
( <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E19:96_1_2_2"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">W</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:96_1_2_2"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">Wj</font>,<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E10:96_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E21:96_1_2_2"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">Wj</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:96_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/><a NAME="E22:96_1_2_2"/><b>then </b>
<font color="Maroon" title="c17">Wj</font>,<font color="Maroon" title="c7">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E23:96_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E13:96_1_2_2"><i><font color="Green" title="E99">A32</font></i></a>, <a class="txt" href="chord.html#E11:96_1_2_2"><i><font color="Green" title="E98">A31</font></i></a>, <a class="txt" href="chord.html#E7:96"><i><font color="Green" title="E84">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:96"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green" title="E86">A33</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E5:96"><i><font color="Green" title="E83">A5</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="E11:96"/><i><font color="Green" title="E87">A34</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E12:96"/><b>then </b><i><font color="Green" title="E88">A35</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="chord.html#E1:96"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="chord.html#E2:96"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E89">A36</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><b>consider </b><font color="Maroon" title="c13">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:96_2"/><i><font color="Green" title="E90">A37</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E1:96"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="chord.html#E2:96"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="chord.html#E1:96_2"><i><font color="Green" title="E89">A36</font></i></a>, <a class="ref" href="chord.html#T71" target="_self" title="CHORD:th.71">Th71</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_1"><b>suppose </b></a><a NAME="E1:96_2_1_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_2_1_1"/>
 ex <font color="Olive" title="b1">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E1:96_2"><i><font color="Green" title="E89">A36</font></i></a>, <a class="txt" href="chord.html#E3:96_2"><i><font color="Green" title="E90">A37</font></i></a>, <a class="txt" href="chord.html#E8:96"><i><font color="Green" title="E85">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_1_2"><b>suppose </b></a><a NAME="E1:96_2_1_2"/><i><font color="Green" title="E91">A38</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">y</font>;<br/><a NAME="E2:96_2_1_2"/>
not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E1:96_2_1_2"><i><font color="Green" title="E91">A38</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:96_2_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E3:96_2"><i><font color="Green" title="E90">A37</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E4:96_2_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E3:96_2"><i><font color="Green" title="E90">A37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E14:96"/><b>then </b>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:96"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="chord.html#E2:96"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="chord.html#E12:96"><i><font color="Green" title="E88">A35</font></i></a>, <a class="ref" href="chord.html#T71" target="_self" title="CHORD:th.71">Th71</a></i>;<br/>
<b>hence </b><a NAME="E15:96"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:96"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="chord.html#E11:96"><i><font color="Green" title="E87">A34</font></i></a>, <a class="txt" href="chord.html#E10:96"><i><font color="Green" title="E86">A33</font></i></a>, <a class="ref" href="chord.html#D9" target="_self" title="CHORD:def.9">Def9</a></i>;<br/>


</div>
