<?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">S</font> be  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>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<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>let </b><font color="Maroon">V</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">H</font>;<br/>









<b>assume </b><a NAME="E1:111"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:111_1"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">W</font> is <a href="chord.html#V5" target="_self">chordal</a>
 ;<br/><b>assume </b><a NAME="E2:111_1"/>
<font color="Maroon">V</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="E4:111_1"/><i><font color="Green">E53</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">V</font> &amp; <font color="Maroon">V</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">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E5:111_1"/><i><font color="Green">E56</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">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:111_1"/><i><font color="Green">E59</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">V</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">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:111"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E7:111_1"/><i><font color="Green">E66</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="E9:111_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E10:111_1"/><i><font color="Green">E68</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>, , <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><a NAME="E11:111_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&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#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:111_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E13:111_1"/><b>then </b>
<font color="Maroon">m</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="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:111_1"/><b>then </b>
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K13">.vertices()</a>  &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><a NAME="E15:111_1"/><b>then </b>
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> )
 ;<br/><a NAME="E16:111_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E17:111_1"/>
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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><b>proof </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="E1:111_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/>

<b>assume </b><a NAME="E2:111_1_1"/><i><font color="Green">E71</font></i>: 
<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>
 ;<br/>

<a NAME="E3:111_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E4:111_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/>
<b>hence </b><a NAME="E5:111_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:111_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E9:111"><i><font color="Green">E80</font></i></a>, , , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:111"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">V</font> is <a href="chord.html#V5" target="_self">chordal</a>
 ;<br/>

<b>assume </b><a NAME="E4:111"/>
<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="E6:111"/><i><font color="Green">E74</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="E7:111"/><i><font color="Green">E75</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="E8:111"/><i><font color="Green">E76</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><a class="txt" href="chord.html#E9:111"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E9:111"/><i><font color="Green">E80</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="E11:111"/><i><font color="Green">E81</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><a class="ref" href="chord.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E12:111"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&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#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:111"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:111"/><b>then </b>
<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>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E15:111"/><b>then </b>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K13">.vertices()</a>  &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K13">.vertices()</a>  )
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<a NAME="E16:111"/><b>then </b>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> )
 
;<br/>
<a NAME="E17:111"/><b>then </b>
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E18:111"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:111"/>
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">V</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">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>proof </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="E1:111_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/>

<b>assume </b><a NAME="E2:111_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 ;<br/>

<a NAME="E3:111_2"/><i><font color="Green">E85</font></i>: 
<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#T75">GLIB_000:75</a></i>;<br/>
<a NAME="E4:111_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a> 
 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/>
<b>hence </b><a NAME="E5:111_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:111"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E9:111"><i><font color="Green">E80</font></i></a>, , , </i>;<br/>


</div>
