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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>given </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><a NAME="E1:100"/><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>
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green">E33</font></i>: 
<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>and </b><br/><a NAME="E3:100"/><i><font color="Green">E53</font></i>: 
<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="E4:100"/><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">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="E5:100"/><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">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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D10" target="_self">CHORD:def 10</a><br/>

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

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

<b>thus </b><a NAME="E6:100"/>
<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>
 <b>by </b><i><a class="txt" href="chord.html#E35:100"><i><font color="Green">E82</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:100"/>
<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="txt" href="chord.html#E36:100"><i><font color="Green">E83</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:100"/>
<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>by </b><i/>;<br/>

<b>thus </b><a NAME="E9:100"/>
 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>by </b><i/>;<br/>

<b>assume </b><a NAME="E10:100"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 ;<br/>

<a NAME="E11:100"/><b>then </b><i><font color="Green">E67</font></i>: 
<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="E12:100"/>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E13:100"/><b>then </b><i><font color="Green">E68</font></i>: 
3 <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="glib_001.html#T126">GLIB_001:126</a></i>;<br/>
<a NAME="E14:100"/><b>then </b><i><font color="Green">E69</font></i>: 
1 <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#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E15:100"/><i><font color="Green">E70</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="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> )
 
;<br/>
<a NAME="E16:100"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>
<b>thus </b><a NAME="E17:100"/>
(  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> )
 <b>by </b><i>, <a class="ref" href="chord.html#T19" target="_self">Th19</a>, , <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/>

<a NAME="E18:100"/>
2 <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/>
<b>then reconsider </b><font color="Maroon">lW2</font> = <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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> 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="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">lW2</font> = <font color="Maroon">lW2</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E21:100"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">lW2</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="E22:100"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lW2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lW2</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lW2</font> <a href="xcmplx_0.html#K2">+</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/>
<a NAME="E23:100"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lW2</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <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> <font color="Maroon">lW2</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>reconsider </b><font color="Maroon">le</font> = <font color="Maroon">lW2</font> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E25:100"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">le</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="E26:100"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">le</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E27:100"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">le</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="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E28:100"/>
<font color="Maroon">le</font> <a href="nat_1.html#K5">div</a> 2 <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#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="glib_001.html#T78">GLIB_001:78</a></i>;<br/>
<a NAME="E29:100"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">le</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E30:100"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <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#T78">GLIB_001:78</a></i>;<br/>
<b>hence </b><a NAME="E31:100"/>
(  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 )
 <b>by </b><i>, </i>;<br/>

<a NAME="E32:100"/><i><font color="Green">E80</font></i>: 
<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 is  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>
 
;<br/>
<a NAME="E33:100"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default">1 <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/>
<a NAME="E34:100"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/>
<a NAME="E35:100"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E36:100"/><i><font color="Green">E83</font></i>: 
1 <a href="nat_1.html#K1">+</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="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E37:100"/>
1 <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#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="glib_001.html#T78">GLIB_001:78</a></i>;<br/>
<a NAME="E38:100"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E39:100"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="glib_001.html#T78">GLIB_001:78</a></i>;<br/>
<b>hence </b><a NAME="E40:100"/>
(  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/>


</div>
