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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">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">U</font> = <font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> ;<br/>
<b>assume </b><a NAME="E1:107"/>
<font color="Maroon">W</font> is <a href="chord.html#V5" target="_self">chordal</a>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:107"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <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> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</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">n</font> )
 <b>and </b><br/><a NAME="E4:107"/><i><font color="Green">E33</font></i>: 
 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">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E5:107"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  holds <br/> not <font color="Olive">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:107"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <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/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:107"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:107"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:107"/><b>then </b>
<font color="Maroon">m</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#T18" target="_self">Th18</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E11:107"/><b>then </b><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; 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">W</font> )
 
<b>by </b><i>, <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E12:107"/><b>then </b><i><font color="Green">E67</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> &amp; <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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">um</font> = <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="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<b>set </b><font color="Maroon">un</font> = <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="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">um</font> = <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="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1, <font color="Maroon">un</font> = <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="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="finseq_5.html#T1">FINSEQ_5:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">um</font> = <font color="Maroon">um</font>, <font color="Maroon">un</font> = <font color="Maroon">un</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E15:107"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">un</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">um</font>
 
<b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E16:107"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:107"/><b>then </b>
<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> <font color="Maroon">m</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#T46">XREAL_1:46</a></i>;<br/>
<a NAME="E18:107"/><b>then </b>
<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="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <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="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E19:107"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">um</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#K6">.reverse()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#T22">GLIB_001:22</a></i>;<br/>
<a NAME="E20:107"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T108">GLIB_001:108</a></i>;<br/>
<a NAME="E21:107"/><i><font color="Green">E71</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#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">um</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">un</font> &amp; <font color="Maroon">un</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> &amp; <font color="Maroon">um</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T25">GLIB_001:25</a></i>;<br/>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>assume </b><a NAME="E2:107_1"/>
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">un</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">um</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E3:107_1"/><b>then </b>
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><b>hence </b><a NAME="E4:107_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E23:107"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">un</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">um</font>,<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_000.html#T17">GLIB_000:17</a></i>;<br/>
<b>hence </b><a NAME="E24:107"/>
<font color="Maroon">W</font> <a href="glib_001.html#K6">.reverse()</a>  is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, , , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, , </i>;<br/>


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