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<div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&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> )
 ;<br/><b>reconsider </b><font color="Maroon">m4</font> = <font color="Maroon">m</font>, <font color="Maroon">n4</font> = <font color="Maroon">n</font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font>;<br/><b>set </b><font color="Maroon">VG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">EG</font> =  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>;<br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&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> )
 <b>by </b><i>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E4:17_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m4</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n4</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m4</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenW2</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></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>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:17_1_1_1_1"/>
not <font color="Maroon">lenW2</font> is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E2:17_1_1_1_1"/>
not  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> is <a href="abian.html#V1">even</a>
 ;<br/><a NAME="E3:17_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenW2aa1</font> = <font color="Maroon">lenW2</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <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/><a NAME="E5:17_1_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">lenW2aa1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:17_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</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> 0</span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E7:17_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <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="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E8:17_1_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K5" target="_self">.vertexAt</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:17_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">i</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E10:17_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span>
 ;<br/><a NAME="E11:17_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:17_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E13:17_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n4</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:17_1_1_1_1"/><b>then </b>
<font color="Maroon">m4</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n4</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E15:17_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:17_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">iaa1</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</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/><a NAME="E18:17_1_1_1_1"/>
<font color="Maroon">iaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E19:17_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">iaa1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</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> <font color="Maroon">iaa1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E20:17_1_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:17_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E22:17_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E23:17_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">m4</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 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/><a NAME="E25:17_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&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> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E26:17_1_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <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> <font color="Maroon">x</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E27:17_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:17_1_1_1"/>
<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:17_1_1"/>
( ( ( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <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> ) implies <font color="Maroon">W</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> holds  verum ) )
 ;<br/>


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