<?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">W</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 ;<br/>

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:37"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font>;<br/>
<a NAME="E4:37"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  )
 
<b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<a NAME="E5:37"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E6:37"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#T65">GLIB_000:65</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">n</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="E3:37_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <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">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>and </b><br/><a NAME="E4:37_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/><a NAME="E5:37_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</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#T41">XREAL_1:41</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  ) or ( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  ) )
 </span><b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1"/>
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:37_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><a NAME="E3:37_1_1_1_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">n</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:37_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, , , <a class="txt" href="chord.html#E2:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:37"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V4">Trail-like</a>
 
<b>by </b><i><a class="ref" href="chord.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_001.html#T143">GLIB_001:143</a></i>;<br/>
<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&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> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:37_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:37_2"/><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E4:37_2"/>
<font color="Maroon">m</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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:37_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&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> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><a NAME="E6:37_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:37_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_1"/>
<font color="Maroon">n</font> <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">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span>
 ;<br/><a NAME="E2:37_2_1"/><b>then </b>
<font color="Maroon">n</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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E3:37_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&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> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><a NAME="E4:37_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a>, , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/><a NAME="E5:37_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E6:37_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:37_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div><a NAME="E9:37_2"/><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#K10">.addEdge</a> <font color="Maroon">e</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> 1
 <b>by </b><i>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:37_2_2"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><a NAME="E3:37_2_2"/><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">m</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:37_2_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/><b>thus </b><a NAME="E5:37_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:37_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font></span>)</span> )
 <b>by </b><i><a class="ref" href="chord.html#T12" target="_self">Th12</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:37"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V5">Path-like</a>
 
<b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/>
<a NAME="E12:37"/><i><font color="Green">E114</font></i>: 
not <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="glib_001.html#T133">GLIB_001:133</a></i>;<br/>
<b>thus </b><a NAME="E13:37"/>
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>


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