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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">G</font>;<br/>



<a NAME="E1:134"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:134"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">V</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1"><b>now </b></a><div class="add"><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>, <font color="Maroon">c</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:134_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/><b>let </b><font color="Maroon">va</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">vb</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">vc</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E2:134_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">va</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">vb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">vc</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">va</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">vb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">vc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vb</font> &amp; <font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vc</font> )
 ;<br/><a NAME="E3:134_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>assume </b><a NAME="E4:134_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">V</font> is <a href="chord.html#V7" target="_self">perfect</a>
 ;<br/><a NAME="E5:134_1"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">va</font> &amp; <font color="Maroon">va</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:134_1"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">va</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:134_1"/><i><font color="Green">E70</font></i>: 
not <font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">va</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, </i>;<br/><b>consider </b><font color="Maroon">Gf</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">va</font></span>)</span>;<br/><b>set </b><font color="Maroon">fs</font> = <font color="Maroon">va</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font>;<br/><a NAME="E9:134_1"/><i><font color="Green">E71</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">va</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">va</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E10:134_1"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">jb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:134_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vb</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>consider </b><font color="Maroon">jc</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:134_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vc</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E15:134_1"/>
<font color="Maroon">va</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font> <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">V</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:134_1"/><b>then </b><i><font color="Green">E75</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T41">XREAL_1:41</a>, </i>;<br/><a NAME="E17:134_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1"/>
<font color="Maroon">jb</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 ;<br/><a NAME="E2:134_1_1"/><b>then </b>
<font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jb</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">va</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:134_1_1"/><b>then </b>
<font color="Maroon">vb</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E38:134_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:134_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><a NAME="E19:134_1"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">jb</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jb</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_2"/>
<font color="Maroon">jc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 ;<br/><a NAME="E2:134_1_2"/><b>then </b>
<font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">va</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">va</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:134_1_2"/><b>then </b>
<font color="Maroon">vc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E39:134_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:134_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><a NAME="E21:134_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">jc</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">va</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jc</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E22:134_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:134_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:134_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">va</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E25:134_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E26:134_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jb</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E27:134_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jb</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">jb</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E28:134_1"/><b>then </b>
<font color="Maroon">jb</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E29:134_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">va</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E38:134_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E30:134_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jc</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E31:134_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jc</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E32:134_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jc</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">jc</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E33:134_1"/><b>then </b>
<font color="Maroon">jc</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:134_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">va</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E39:134_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">ag</font> = <font color="Maroon">a</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a>, <a class="ref" href="chord.html#T27" target="_self">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon">bg</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a>, </i>;<br/><b>reconsider </b><font color="Maroon">cg</font> = <font color="Maroon">c</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a>, </i>;<br/><a NAME="E38:134_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">ag</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, , , </i>;<br/><a NAME="E39:134_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">ag</font>,<font color="Maroon">bg</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">ag</font>,<font color="Maroon">cg</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; not <font color="Maroon">bg</font>,<font color="Maroon">cg</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E40:134_1"/><b>then </b>
( <font color="Maroon">bg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gf</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ag</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">cg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gf</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ag</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E41:134_1"/><b>then </b>
 ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">bg</font>,<font color="Maroon">cg</font>,<font color="Maroon">Gf</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/><b>hence </b><a NAME="E42:134_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2"/>
not <font color="Maroon">V</font> is <a href="chord.html#V7" target="_self">perfect</a>
 ;<br/><a NAME="E2:134_2"/><b>then </b>
 ex <font color="Olive">n</font> being non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> &amp;  ex <font color="Olive">Gf</font> being  <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Olive">n</font></span>)</span> ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gf</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; not <font color="Olive">v</font> is <a href="chord.html#V3" target="_self">simplicial</a> ) )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">n</font> being  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">Gf</font> being   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font><b> such that </b><br/><a NAME="E4:134_2"/><i><font color="Green">E114</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">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a> )
 ;<br/><a NAME="E5:134_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E6:134_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font>, <font color="Maroon">b</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font><b> such that </b><br/><a NAME="E8:134_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>and </b><br/><a NAME="E9:134_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">a</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E10:134_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E11:134_2"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Gf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>reconsider </b><font color="Maroon">vg</font> = <font color="Maroon">v</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E13:134_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">ag</font> = <font color="Maroon">a</font>, <font color="Maroon">bg</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E15:134_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E16:134_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">vg</font>,<font color="Maroon">ag</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">vg</font>,<font color="Maroon">bg</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; not <font color="Maroon">ag</font>,<font color="Maroon">bg</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E17:134_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="chord.html#T21" target="_self">Th21</a></i>;<br/><b>consider </b><font color="Maroon">va</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:134_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">va</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">va</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E20:134_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="chord.html#T21" target="_self">Th21</a></i>;<br/><b>consider </b><font color="Maroon">vb</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:134_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">vb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">vb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2_1"/>
<font color="Maroon">va</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:134_2_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:134_2_1"/>
<font color="Maroon">a</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="chord.html#T25" target="_self">Th25</a>, , <a class="ref" href="chord.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E4:134_2_1"/><b>then </b>
<font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E5:134_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E198</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2_2"/>
<font color="Maroon">vb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:134_2_2"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">vb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="txt" href="chord.html#E6:134_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:134_2_2"/>
<font color="Maroon">b</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E4:134_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="chord.html#T25" target="_self">Th25</a>, , <a class="ref" href="chord.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E4:134_2_2"/><b>then </b>
<font color="Maroon">vb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:134_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E5:134_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E6:134_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_4.html#T34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_2_3"/>
( <font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vb</font> or <font color="Maroon">vb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">va</font> )
 </span><b>by </b><i>, , <a class="txt" href="chord.html#E6:134_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3_1"><b>suppose </b></a><a NAME="E1:134_2_3_1"/>
<font color="Maroon">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">vb</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:134_2_3_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ex <font color="Olive">va</font>, <font color="Olive">vb</font>, <font color="Olive">vc</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">va</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vc</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">va</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vb</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vc</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vb</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vc</font> &amp; not <font color="Olive">b</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, , <a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="txt" href="chord.html#E6:134_2"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="chord.html#E8:134_2"><i><font color="Green">E117</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3_2"><b>suppose </b></a><a NAME="E1:134_2_3_2"/>
<font color="Maroon">vb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">va</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:134_2_3_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ex <font color="Olive">va</font>, <font color="Olive">vb</font>, <font color="Olive">vc</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">va</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vc</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">va</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vb</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vc</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vb</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vc</font> &amp; not <font color="Olive">b</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, , <a class="ref" href="chord.html#T24" target="_self">Th24</a>, , <a class="txt" href="chord.html#E6:134_2"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="chord.html#E8:134_2"><i><font color="Green">E117</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:134"/>
( <font color="Maroon">V</font> is <a href="chord.html#V7" target="_self">perfect</a> iff for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  holds <br/>for <font color="Olive">va</font>, <font color="Olive">vb</font>, <font color="Olive">vc</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">va</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Olive">vc</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">va</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vb</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">vc</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vb</font> &amp; <font color="Olive">va</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">vc</font> holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a></i>;<br/>


</div>
