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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>assume </b><b>that </b><br/><a NAME="E1:104"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a> &amp; <font color="Maroon">W</font> is <a href="chord.html#V5" target="_self">chordal</a> )
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 4
 ;<br/>

<a NAME="E3:104"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E4:104"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E5:104"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</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>assume </b><a NAME="E6:104"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( not <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> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 5,<font color="Maroon">G</font> &amp; not <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> 3,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 7,<font color="Maroon">G</font> )
 ;<br/>

<b>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="E8:104"/><i><font color="Green">E66</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> &amp;  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> &amp; ( <font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a> implies ( (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <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="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> ) ) ) )
 <b>by </b><i>, </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="E10:104"/><i><font color="Green">E67</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="E11:104"/><i><font color="Green">E68</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:104_1"/><b>then </b>
<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="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:104_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 9 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E5:104_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">jj</font>
 <b>by </b><i>, , , , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:104_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</font> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><a NAME="E7:104_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 5
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , </i>;<br/><a NAME="E8:104_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">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:104_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:104_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E13:104"/><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="E14:104"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><a NAME="E2:104_2"/><b>then </b>
<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="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:104_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 9 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i>, </i>;<br/><a NAME="E4:104_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 7
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, , </i>;<br/><a NAME="E5:104_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:104_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 7 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:104_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E16:104"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</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="E17:104"/><b>then </b><i><font color="Green">E75</font></i>: 
3 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 9
 ;<br/><a NAME="E2:104_3_1"/><b>then </b>
<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> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 5,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/><b>hence </b><a NAME="E3:104_3_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:104_3"/><b>then </b>
<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="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:104_3"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&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="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:104_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E19:104"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</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="E20:104"/><b>then </b>
5 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:104"/><b>then </b>
7 <a href="nat_1.html#K1">+</a> 2 <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/>
<b>hence </b><a NAME="E22:104"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


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