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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">P</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:109"/><i><font color="Green">E30</font></i>: 
( not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a> )
 ;<br/>

<b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:109"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">m</font> <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">P</font> )
 ;<br/>

<a NAME="E3:109"/><i><font color="Green">E53</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/>
<a NAME="E4:109"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">P</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">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T148">GLIB_001:148</a></i>;<br/>
<a NAME="E5:109"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V6">vertex-distinct</a>
 
<b>by </b><i>, <a class="txt" href="chord.html#E1:109"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:109"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:109_1"/><i><font color="Green">E67</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">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:109_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_1"/><i><font color="Green">E69</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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_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="E1:109_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>consider </b><font color="Maroon">k</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> <b> such that </b><br/><a NAME="E3:109_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E4:109_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:109_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><b>assume </b><a NAME="E6:109_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_1_1_1"/>
( ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> ) or ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) )
 </span><b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_1"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> )
 <b>by </b><i>, , , , , , <a class="ref" href="glib_001.html#D29">GLIB_001:def 29</a></i>;<br/><b>hence </b><a NAME="E3:109_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_2"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="glib_001.html#D29">GLIB_001:def 29</a></i>;<br/><a NAME="E3:109_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:109_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:109_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="chord.html#T19" target="_self">Th19</a>, , , <a class="txt" href="chord.html#E8:109"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:109_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:109"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<b>take </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
;<br/>

<a NAME="E9:109"/>
<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">P</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E10:109"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</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_001.html#D3">GLIB_001:def 3</a></i>;<br/>


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