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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:180_1_1"/>
( ( not <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> ) or <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_1_1"><b>suppose </b></a><a NAME="E1:180_1_1_1"/><i><font color="Green">E128</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:180_1_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:180_1_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:180_1_1_1_1"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:180_1_1_1_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="glib_001.html#E3:180_1_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:180_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E4:180_1_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:180_1_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:180_1_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="glib_001.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E8:180_1_1_1_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:180_1_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="glib_001.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E9:180_1_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:180_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="glib_001.html#E7:180_1_1_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="glib_001.html#E64"><i><font color="Green">Lemma56</font></i></a></i>;<br/><a NAME="E10:180_1_1_1_1"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E8:180_1_1_1_1"><i><font color="Green">E133</font></i></a></i>;<br/><a NAME="E11:180_1_1_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <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">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E8:180_1_1_1_1"><i><font color="Green">E133</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:180_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:180_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E13:180_1_1_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:180_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:180_1_1_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E10:180_1_1_1_1"><i><font color="Green">E135</font></i></a>, <a class="txt" href="glib_001.html#E11:180_1_1_1_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="glib_001.html#E13:180_1_1_1_1"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:180_1_1_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E16:180_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:180_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:180_1_1_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="glib_001.html#D14" target="_self">Def14</a></i>;<br/><b>hence </b><a NAME="E18:180_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:180_1_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E15:180_1_1_1_1"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:180_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_1_2"><b>suppose </b></a><a NAME="E1:180_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:180_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:180"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> 
 ;<br/>


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