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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="xxreal_0.html#T1">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">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="graph_2.html#T7">GRAPH_2:7</a></i>;<br/><a NAME="E3:14_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:14_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E5:14_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E6:14_1_1"/><b>then </b>
1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E7:14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T47">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">E11</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:14_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b>
<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">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E4:14_1_2"/><b>then </b>
 <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="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E5:14_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="graph_2.html#T7">GRAPH_2:7</a></i>
;<br/><b>hence </b><a NAME="E7:14_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></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">E11</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:14_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:14_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:14_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; 1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="graph_2.html#D1">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">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">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">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="graph_1.html#T14">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">E11</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E2:14_1_4"/><i><font color="Green">E12</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self">Th8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:14_1_4"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36">GRAPH_2:36</a></i>;<br/><a NAME="E6:14_1_4"/><i><font color="Green">E14</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/><a NAME="E7:14_1_4"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msscyc_1.html#T6">MSSCYC_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E9:14_1_4"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><a NAME="E10:14_1_4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="graph_3.html#E5:14_1_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="graph_3.html#E6:14_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><a NAME="E11:14_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:14_1_4"/><b>then </b><i><font color="Green">E18</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="graph_3.html#E6:14_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:14_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:14_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:14_1_4"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E2:14_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E16:14_1_4"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E17:14_1_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E18:14_1_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E19:14_1_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E20:14_1_4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E21:14_1_4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="graph_2.html#T9">GRAPH_2:9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="graph_2.html#T46">GRAPH_2:46</a></i>;<br/><a NAME="E24:14_1_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/><a NAME="E25:14_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E21:14_1_4"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E7:14_1_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graph_3.html#E9:14_1_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E10:14_1_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E17:14_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_3.html#E20:14_1_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E27:14_1_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</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">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E12:14_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E28:14_1_4"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E15:14_1_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E29:14_1_4"/><b>then </b><i><font color="Green">E27</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:14_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E16:14_1_4"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E31:14_1_4"/><b>then </b><i><font color="Green">E28</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:14_1_4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E18:14_1_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="graph_3.html#E19:14_1_4"><i><font color="Green">E23</font></i></a>, <a class="txt" href="graph_3.html#E29:14_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E31:14_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>
;<br/><a NAME="E33:14_1_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E24:14_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/><b>hence </b><a NAME="E34:14_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
