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







<b>assume </b><a NAME="E1:46"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> is <a href="chord.html#V1" target="_self">minlength</a>
 ;<br/>

<a NAME="E2:46"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">GA</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><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="E1:46_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">P</font> )
 ;<br/><a NAME="E2:46_1"/><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>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:46_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">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="E4:46_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="E5:46_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="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:46_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">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:46_1"/><b>then </b>
( <font color="Maroon">P</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">GA</font> &amp; <font color="Maroon">P</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">GA</font> )
 <b>by </b><i><a class="ref" href="chord.html#T6" target="_self">Th6</a>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E8:46_1"/><b>then </b>
<font color="Maroon">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">GA</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a>, <a class="ref" href="chord.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E9:46_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
for <font color="Olive">m</font>, <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a> <a href="abian.html#V1">odd</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> holds <br/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <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="Olive">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">G</font>
 ;<br/>


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