<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">v</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:78_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E2:78_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:78_1_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:78_1_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Rotated2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c9</font> where <font color="Olive" title="b1">c9</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font> &amp;  ex <font color="Olive" title="b2">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c9</font> &amp; <font color="Olive" title="b2">vs</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="c5">v</font> ) ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="graph_3.html#T57" target="_self" title="GRAPH_3:th.57">Th57</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">r2</font> =  <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font>;<br/>
<a NAME="E7:78_1_1"/>
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c6">Rotated2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="graph_3.html#D9" target="_self" title="GRAPH_3:def.9">Def9</a></span>;<br/>
<a NAME="E8:78_1_1"/><span class="kw">then </span>
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Rotated2</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">c99</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <span class="kw"> such that </span><br/><a NAME="E10:78_1_1"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c99</font>
 <span class="kw">and </span><br/><a NAME="E11:78_1_1"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font>
 <span class="kw">and </span><br/><a NAME="E12:78_1_1"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
 ex <font color="Olive" title="b1">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c8">c99</font> &amp; <font color="Olive" title="b1">vs</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="c5">v</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">vs2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E14:78_1_1"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c9">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c8">c99</font>
 <span class="kw">and </span><br/><a NAME="E15:78_1_1"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c9">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="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:78_1_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">c92</font> = <font color="Maroon" title="c8">c99</font> as   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/>
<a NAME="E17:78_1_1"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c92</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:78_1_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a></span>;<br/>
<a NAME="E18:78_1_1"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="graph_3.html#T21" target="_self" title="GRAPH_3:th.21">Th21</a></span>;<br/>
<a NAME="E19:78_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c8">c99</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:78_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a></span>;<br/>
<a NAME="E20:78_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c8">c99</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E21:78_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
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="c9">vs2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:78_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></span>;<br/>
<a NAME="E22:78_1_1"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
<font color="Maroon" title="c9">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c10">c92</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:78_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Rotated1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c9</font> where <font color="Olive" title="b1">c9</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  : (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font> &amp;  ex <font color="Olive" title="b2">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c9</font> &amp; <font color="Olive" title="b2">vs</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="c5">v</font> ) ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="graph_3.html#T57" target="_self" title="GRAPH_3:th.57">Th57</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">r1</font> =  <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font>;<br/>
<a NAME="E24:78_1_1"/>
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c11">Rotated1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="graph_3.html#D9" target="_self" title="GRAPH_3:def.9">Def9</a></span>;<br/>
<a NAME="E25:78_1_1"/><span class="kw">then </span>
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Rotated1</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">c9</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> <span class="kw"> such that </span><br/><a NAME="E27:78_1_1"/><span class="lab"><font color="Green" title="E74">A15</font></span>: 
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c9</font>
 <span class="kw">and </span><br/><a NAME="E28:78_1_1"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font>
 <span class="kw">and </span><br/><a NAME="E29:78_1_1"/><span class="lab"><font color="Green" title="E76">A17</font></span>: 
 ex <font color="Olive" title="b1">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c13">c9</font> &amp; <font color="Olive" title="b1">vs</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="c5">v</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">c91</font> = <font color="Maroon" title="c13">c9</font> as   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">vs1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E32:78_1_1"/><span class="lab"><font color="Green" title="E77">A18</font></span>: 
<font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c13">c9</font>
 <span class="kw">and </span><br/><a NAME="E33:78_1_1"/><span class="lab"><font color="Green" title="E78">A19</font></span>: 
<font color="Maroon" title="c15">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="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:78_1_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a></span>;<br/>
<a NAME="E34:78_1_1"/>
<font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c14">c91</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:78_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a></span>;<br/>
<a NAME="E35:78_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<font color="Maroon" title="c15">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="c15">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="c15">vs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">vs12</font> = <font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs2</font>;<br/>
<a NAME="E36:78_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs1</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="c13">c9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:78_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></span>;<br/>
<a NAME="E37:78_1_1"/><span class="kw">then </span>
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="c15">vs1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E80">A21</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c15">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></span>
<br/>.= 
<font color="Maroon" title="c9">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="c9">vs2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:78_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="txt" href="#E15:78_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E22:78_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs2</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:78_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></span>
;<br/>
<a NAME="E39:78_1_1"/><span class="lab"><font color="Green" title="E81">A22</font></span>: 
 <a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c91</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:78_1_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">c</font> = <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font></span>)</span> as   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E28:78_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E11:78_1_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E32:78_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E33:78_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="txt" href="#E14:78_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E15:78_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E35:78_1_1"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="txt" href="#E17:78_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="graph_3.html#T9" target="_self" title="GRAPH_3:th.9">Th9</a></span>;<br/>
<a NAME="E41:78_1_1"/>
<font color="Maroon" title="c15">vs1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c17">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:78_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E33:78_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="txt" href="#E14:78_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E15:78_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E35:78_1_1"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="txt" href="#E39:78_1_1"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="txt" href="#E17:78_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></span>;<br/>
<a NAME="E42:78_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">c</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:78_1_1"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">c</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Olive" title="b1">v</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Olive" title="b1">v</font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E44:78_1_1"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c5">v</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
