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<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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><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> )
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v1</font>,<font color="Maroon">v2</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v2</font>,<font color="Maroon">v3</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>consider </b><font color="Maroon">e1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:53"/><i><font color="Green">E59</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>
 <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">e2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:53"/><i><font color="Green">E66</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">P1</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font>;<br/>
<a NAME="E9:53"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>;<br/>
<a NAME="E12:53"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T92">GLIB_001:92</a></i>;<br/>
<a NAME="E13:53"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i>, , <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<b>set </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font>;<br/>
<a NAME="E14:53"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<a NAME="E15:53"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/>
<a NAME="E16:53"/>
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K13">.vertices()</a> 
 
<b>by </b><i>, <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <font color="Maroon">e2</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, , , <a class="ref" href="glib_001.html#T152">GLIB_001:152</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>take </b>
<font color="Maroon">e1</font>
;<br/>

<b>take </b>
<font color="Maroon">e2</font>
;<br/>

<b>thus </b><a NAME="E18:53"/>
not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E4:53"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>

<a NAME="E19:53"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, , <a class="txt" href="chord.html#E3:53"><i><font color="Green">E53</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/>
<b>hence </b><a NAME="E20:53"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 5
 ;<br/>

<a NAME="E21:53"/>
5 <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="E22:53"/>
<font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/>

<b>thus </b><a NAME="E23:53"/>
<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>
 <b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a></i>;<br/>

<b>thus </b><a NAME="E24:53"/>
<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>
 <b>by </b><i/>;<br/>

<a NAME="E25:53"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T109">GLIB_001:109</a></i>;<br/>
<a NAME="E26:53"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e1</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T112">GLIB_001:112</a></i>;<br/>
<b>hence </b><a NAME="E27:53"/>
<font color="Maroon">P</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>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>

<a NAME="E28:53"/>
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/>
<b>hence </b><a NAME="E29:53"/>
<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>
 <b>by </b><i><a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>

<b>thus </b><a NAME="E30:53"/>
<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#E4:53"><i><font color="Green">E56</font></i></a></i>;<br/>

<a NAME="E31:53"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="chord.html#E3:53"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 3 = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">v1</font>,<font color="Maroon">e1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> 3
<b>by </b><i>, , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i>, <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#T15">GLIB_001:15</a></i>
;<br/><br/>

<b>thus </b><a NAME="E33:53"/>
<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>, </i>;<br/>


</div>
