<?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">v1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v3</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v4</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="E1:54"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v4</font> &amp; <font color="Maroon">v3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v4</font> )
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v1</font>,<font color="Maroon">v2</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">v2</font>,<font color="Maroon">v3</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  &amp; <font color="Maroon">v3</font>,<font color="Maroon">v4</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">e3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e3</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v3</font>,<font color="Maroon">v4</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T12" target="_self">Th12</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <font color="Maroon">e1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:54"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">R</font> is <a href="glib_001.html#V1">closed</a>
 <b>and </b><br/><a NAME="E7:54"/><i><font color="Green">E59</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 5 &amp; <font color="Maroon">R</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 2 )
 <b>and </b><br/><a NAME="E8:54"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">e1</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v2</font>,<font color="Maroon">v3</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>and </b><br/><a NAME="E9:54"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">R</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="enumset1.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E10:54"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> )
 <b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="chord.html#T12" target="_self">Th12</a>, <a class="ref" href="chord.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font>
 
<b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, , <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/>
<a NAME="E12:54"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e3</font>
 
<b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, , <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/>
<a NAME="E13:54"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">e3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E14:54"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">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>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54_1"/><i><font color="Green">E73</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:54_1"/><i><font color="Green">E74</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">R</font>
 ;<br/>

<a NAME="E3:54_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 6
 
<b>by </b><i>, <a class="txt" href="chord.html#E7:54"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1"/>
<font color="Maroon">R</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">v4</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_2"/>
<font color="Maroon">R</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">v4</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e3</font>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e3</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="txt" href="chord.html#E6:54"><i><font color="Green">E56</font></i></a>, <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<a NAME="E16:54"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 5 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<b>hence </b><a NAME="E17:54"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 7
 ;<br/>

<a NAME="E18:54"/>
7 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/>
<b>hence </b><a NAME="E19:54"/>
<font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 3
 ;<br/>

<a NAME="E20:54"/>
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v4</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/>
<b>hence </b><a NAME="E21:54"/>
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font>,<font color="Maroon">v4</font></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T46">ENUMSET1:46</a></i>;<br/>

<a NAME="E22:54"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E23:54"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/>

<a NAME="E24:54"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E25:54"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/>

<a NAME="E26:54"/>
5 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E27:54"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 5 <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/>

<a NAME="E28:54"/>
<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v4</font>
 
<b>by </b><i>, <a class="txt" href="chord.html#E4:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<b>hence </b><a NAME="E29:54"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 7 <a href="hidden.html#R1">=</a> <font color="Maroon">v4</font>
 <b>by </b><i/>;<br/>


</div>
