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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> ) )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T7" title="GRAPH_2:th.7">GRAPH_2:7</a></i>;<br/><a NAME="E3:14_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 ;<br/><a NAME="E4:14_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E5:14_1_1"/><b>then </b>
1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E6:14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E4:14_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> = 
1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="graph_2.html#T7" title="GRAPH_2:th.7">GRAPH_2:7</a></i>
;<br/><b>hence </b><a NAME="E6:14_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_3"><b>suppose </b></a><a NAME="E1:14_1_3"/><i><font color="Green" title="E11">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_3"/>
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:14_1_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_3"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:14_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_3"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><a NAME="E5:14_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E6:14_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T12" target="_self" title="GRAPH_3:th.12">Th12</a>, <a class="ref" href="graph_1.html#T14" title="GRAPH_1:th.14">GRAPH_1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_4"><b>suppose </b></a><a NAME="E1:14_1_4"/><i><font color="Green" title="E11">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c2">m</font>;<br/><b>set </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:14_1_4"/><i><font color="Green" title="E12">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">r1</font> = 1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c7">r2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self" title="GRAPH_3:th.8">Th8</a></i>;<br/><b>set </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r1</font>;<br/><b>consider </b><font color="Maroon" title="c9">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:14_1_4"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c9">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36" title="GRAPH_2:th.36">GRAPH_2:36</a></i>;<br/><a NAME="E6:14_1_4"/><i><font color="Green" title="E14">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/><a NAME="E7:14_1_4"/><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">vs1</font> = 1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c9">vs</font>, <font color="Maroon" title="c11">vs2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c9">vs</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E9:14_1_4"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c10">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c6">r1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="graph_2.html#T45" title="GRAPH_2:th.45">GRAPH_2:45</a></i>;<br/><a NAME="E10:14_1_4"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c11">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c7">r2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="graph_3.html#E6:14_1_4"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="graph_2.html#T45" title="GRAPH_2:th.45">GRAPH_2:45</a></i>;<br/><a NAME="E11:14_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:14_1_4"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="graph_3.html#E6:14_1_4"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:14_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E14:14_1_4"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:14_1_4"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E16:14_1_4"/><i><font color="Green" title="E20">A14</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">vs1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><a NAME="E17:14_1_4"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c10">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E18:14_1_4"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c10">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E19:14_1_4"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c11">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E20:14_1_4"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c11">vs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></i>;<br/><a NAME="E21:14_1_4"/><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">r2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="graph_2.html#T9" title="GRAPH_2:th.9">GRAPH_2:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">vs'</font> = <font color="Maroon" title="c11">vs2</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c10">vs1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>reconsider </b><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c7">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r1</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="graph_2.html#T46" title="GRAPH_2:th.46">GRAPH_2:46</a></i>;<br/><a NAME="E24:14_1_4"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c12">vs'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c13">r</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></i>;<br/><a NAME="E25:14_1_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">r1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">r2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E21:14_1_4"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> as   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self" title="GRAPH_3:th.9">Th9</a></i>;<br/><a NAME="E27:14_1_4"/>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E28:14_1_4"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E15:14_1_4"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E29:14_1_4"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">vs2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E30:14_1_4"/><i><font color="Green" title="E28">A22</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">vs1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="graph_3.html#E16:14_1_4"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><font color="Maroon" title="c12">vs'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c10">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">vs1</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E18:14_1_4"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="graph_3.html#E19:14_1_4"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="graph_3.html#E29:14_1_4"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>
<br/>.= 
<font color="Maroon" title="c12">vs'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs'</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E30:14_1_4"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></i>
;<br/><a NAME="E32:14_1_4"/><b>then </b>
<font color="Maroon" title="c14">r</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E24:14_1_4"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></i>;<br/><b>hence </b><a NAME="E33:14_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon" title="c2">m</font> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c3">p</font></span>)</span> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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