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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <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/>  for <font color="Olive" title="b2">p</font> being   <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> holds  <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="Olive" title="b2">p</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Olive" title="b2">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="Olive" title="b1">m</font>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Olive" title="b2">p</font></span>)</span> is   <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><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <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> holds  <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="Olive" title="b1">p</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Olive" title="b1">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="Olive" title="b1">p</font></span>)</span> is   <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><br/><span class="kw">let </span><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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:13_1"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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><span class="kw">by</span> <span class="lab"><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></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><span class="kw">suppose </span></a><a NAME="E1:13_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:13_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="E3:13_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="#E1:13_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:13_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:13_1_1"/>
(<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="graph_2.html#T7" title="GRAPH_2:th.7">GRAPH_2:7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:13_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#K2" title="NAT_1:func.2">+</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#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="#E4:13_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2"><span class="kw">suppose </span></a><a NAME="E1:13_1_2"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:13_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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E3:13_1_2"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:13_1_2"/><span class="kw">then </span>
(<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="graph_2.html#T7" title="GRAPH_2:th.7">GRAPH_2:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:13_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#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_3"><span class="kw">suppose </span></a><a NAME="E1:13_1_3"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:13_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#K2" title="NAT_1:func.2">+</a> 1
 <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/><a NAME="E3:13_1_3"/><span class="kw">then </span>
 <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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_3"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:13_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</font></span>: 
(<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E5:13_1_3"/>
(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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_3"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E6:13_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1_3"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E7:13_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#K2" title="NAT_1:func.2">+</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#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#T7" target="_self" title="GRAPH_3:th.7">Th12</a>, <a class="ref" href="graph_1.html#T14" title="GRAPH_1:th.14">GRAPH_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_4"><span class="kw">suppose </span></a><a NAME="E1:13_1_4"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c2">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E2:13_1_4"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_4"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><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#K2" title="NAT_1:func.2">+</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#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#T5" target="_self" title="GRAPH_3:th.5">Th8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E5:13_1_4"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T33" title="GRAPH_2:th.33">GRAPH_2:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">vs1</font> = (1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c8">vs</font>, <font color="Maroon" title="c10">vs2</font> = (<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c8">vs</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c8">vs</font> as    <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> ;<br/><a NAME="E7:13_1_4"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_4"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:13_1_4"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_4"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="graph_2.html#D6" title="GRAPH_2:def.6">GRAPH_2:def 6</a></span>;<br/><a NAME="E9:13_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_4"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E7:13_1_4"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="txt" href="#E5:13_1_4"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="graph_2.html#T42" title="GRAPH_2:th.42">GRAPH_2:42</a></span>;<br/><a NAME="E10:13_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#K2" title="NAT_1:func.2">+</a> 1
 <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/><a NAME="E11:13_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c8">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1_4"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="txt" href="#E8:13_1_4"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:13_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A13</font></span>: 
<font color="Maroon" title="c2">m</font> <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">vs</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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:13_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">vs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1</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="#E2:13_1_4"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E11:13_1_4"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E14:13_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A14</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">vs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_4"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:13_1_4"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_4"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E5:13_1_4"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="graph_2.html#T42" title="GRAPH_2:th.42">GRAPH_2:42</a></span>;<br/><a NAME="E16:13_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">vs</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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E17:13_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c8">vs</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="#E11:13_1_4"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:13_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">vs2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c8">vs</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="#E2:13_1_4"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E19:13_1_4"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c10">vs2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:13_1_4"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E20:13_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A16</font></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="c10">vs2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">vs9</font> = <font color="Maroon" title="c10">vs2</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c9">vs1</font> as    <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> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">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/><a NAME="E22:13_1_4"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c8">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="c8">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_4"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></span>;<br/><a NAME="E23:13_1_4"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
( <font color="Maroon" title="c9">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="c8">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c10">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="c10">vs2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c8">vs</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_4"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E11:13_1_4"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:13_1_4"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E15:13_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E9:13_1_4"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="graph_2.html#T43" title="GRAPH_2:th.43">GRAPH_2:43</a></span>;<br/><a NAME="E25:13_1_4"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<font color="Maroon" title="c11">vs9</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:13_1_4"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E15:13_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E9:13_1_4"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E23:13_1_4"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="graph_2.html#T44" title="GRAPH_2:th.44">GRAPH_2:44</a></span>;<br/><a NAME="E26:13_1_4"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_4"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="graph_2.html#T9" title="GRAPH_2:th.9">GRAPH_2:9</a></span>;<br/><a NAME="E27:13_1_4"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">r</font> 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="#E22:13_1_4"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E15:13_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E9:13_1_4"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E23:13_1_4"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="graph_3.html#T6" target="_self" title="GRAPH_3:th.6">Th9</a></span>;<br/><a NAME="E29:13_1_4"/>
( <font color="Maroon" title="c9">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="c9">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">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="c8">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_4"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E11:13_1_4"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="graph_2.html#T12" title="GRAPH_2:th.12">GRAPH_2:12</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c9">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="c9">vs1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:13_1_4"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></span>
<br/>.= 
<font color="Maroon" title="c11">vs9</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">vs9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1_4"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></span>
;<br/><a NAME="E31:13_1_4"/><span class="kw">then </span>
<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#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="#E25:13_1_4"><span class="lab"><font color="Green" title="E23">A19</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">hence </span><a NAME="E32:13_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#K2" title="NAT_1:func.2">+</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#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
