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

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:96"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:96"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>assume </b><a NAME="E3:96"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">aa</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/>



<b>assume </b><a NAME="E4:96"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E5:96"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font> <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">T</font> = <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">aa</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">bb</font>;<br/>
<b>assume </b><a NAME="E7:96"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">aa</font> or  not <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/>

<a NAME="E8:96"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>  st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> <a href="glib_001.html#K13">.vertices()</a>  holds <br/> ex <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> <a href="glib_001.html#K13">.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">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:96_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:96_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:96_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<a NAME="E5:96_1"/><i><font color="Green">E71</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">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">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:96_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> 
 ;<br/><a NAME="E3:96_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E4:96_1_1"/><b>then </b>
not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E7:96"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E5:96_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D19">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">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">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">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:96_1"/><b>then </b><i><font color="Green">E72</font></i>: 
3 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:96_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E11:96_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E13:96_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T116">GLIB_001:116</a></i>;<br/>
<a NAME="E14:96_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E15:96_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">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">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</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">E76</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:96_1_2_1"/>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="glib_001.html#T8">GLIB_001:8</a>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E3:96_1_2_1"/>
 ex <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><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">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:96_1_2_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 ;<br/><b>set </b><font color="Maroon">P</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font>;<br/><a NAME="E3:96_1_2_2"/><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:96_1_2_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="glib_001.html#T37">GLIB_001:37</a></i>;<br/><a NAME="E5:96_1_2_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> holds <br/>( not <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">n</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:96_1_2_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:96_1_2_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E3:96_1_2_2_1"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:96_1_2_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">nu1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E6:96_1_2_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:96_1_2_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E8:96_1_2_2_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nu1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">nu1</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T37">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">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:96_1_2_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:96_1_2_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E4:96_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="glib_001.html#T95">GLIB_001:95</a></i>;<br/><a NAME="E5:96_1_2_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E6:96_1_2_2_1_1"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:96_1_2_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:96_1_2_2_1_1"/>
<font color="Maroon">S</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E8:96_1_2_2_1_1"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E9:96_1_2_2_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:96_1_2_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:96_1_2_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:96_1_2_2_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E13:96_1_2_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D1">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="ref" href="chord.html#T23" target="_self">Th23</a>, , <a class="ref" href="glib_001.html#T116">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">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:96_1_2_2"/><b>then </b>
for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> holds <br/> not <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">HP</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">H</font> <b>by </b><i>, </i>;<br/><a NAME="E8:96_1_2_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">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">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:96_1_2_2"/><b>then </b>
<font color="Maroon">aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HP</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E4:96_1_2_2"><i><font color="Green">E82</font></i></a>, , , <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E11:96_1_2_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">HP</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="glib_002.html#K1">.reachableFrom</a> <font color="Maroon">aa</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T13">GLIB_002:13</a></i>;<br/><a NAME="E12:96_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HP</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E13:96_1_2_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HP</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i/>;<br/><a NAME="E14:96_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">Wj</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E16:96_1_2_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:96_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K16">.find</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:96_1_2_2"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E19:96_1_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E20:96_1_2_2"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Wj</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E21:96_1_2_2"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Wj</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D19">GLIB_001:def 19</a></i>;<br/><a NAME="E22:96_1_2_2"/><b>then </b>
<font color="Maroon">Wj</font>,<font color="Maroon">x</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E23:96_1_2_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:96"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:96"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E12:96"/><i><font color="Green">E126</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="txt" href="chord.html#E7:96"><i><font color="Green">E66</font></i></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">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:96_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:96_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i>, , , </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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_2_1_1"/>
 ex <font color="Olive">y</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i>, , </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">E129</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><a NAME="E2:96_2_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:96_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E4:96_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><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">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E15:96"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, , </i>;<br/>


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