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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" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>set </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:54"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">m</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> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1"/>
( ( not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &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> ) or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> 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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1"/><i><font color="Green">E40</font></i>: 
( not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &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/><div class="add"><a NAME="E2:54_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; 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="heyting3.html#T1">HEYTING3:1</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:54_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E11:54_1_1_1"><i><font color="Green">E48</font></i></a>, , </i>;<br/><a NAME="E4:54_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E5:54_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1,<font color="Maroon">m</font> <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, </i>;<br/><a NAME="E6:54_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#T3" target="_self">Th3</a>, <a class="txt" href="glib_001.html#E11:54_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E7:54_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E8:54_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E11:54_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E9:54_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E10:54_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1,<font color="Maroon">n</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, </i>;<br/><a NAME="E11:54_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1,<font color="Maroon">m</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, </i>;<br/><a NAME="E12:54_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E11:54_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:54_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:54_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">xaa1</font> = <font color="Maroon">x</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="E4:54_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:54_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">xaa1</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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="glib_001.html#T3" target="_self">Th3</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:54_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">xaa1</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"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</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="E7:54_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E8:54_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">xaa1</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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</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="E9:54_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">xaa1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_001.html#T3" target="_self">Th3</a>, , , <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<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> <font color="Maroon">xaa1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="glib_001.html#D1" target="_self">Def1</a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a>, <a class="txt" href="glib_001.html#E8:54_1_1_1_1"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="glib_001.html#E8:54_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_001.html#T5" target="_self">Th5</a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E14:54_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2"/>
( <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> 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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
 ;<br/>


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