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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] </span>}</span> ;<br/>
<a NAME="E1:69_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:69_1_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] </span>}</span>  holds <br/><font color="Olive">x</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] </span>}</span>  as    <a href="finseq_2.html#M1">FinSequenceSet</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#D3">FINSEQ_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_1.html#T14">GRAPH_1:14</a></i>;<br/>
<a NAME="E5:69_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E7:69_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="graph_2.html#T35">GRAPH_2:35</a></i>;<br/>
<a NAME="E8:69_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E9:69_1_1"/><b>then </b>
<font color="Maroon">p</font> is <a href="msscyc_1.html#V1">cyclic</a>
 
<b>by </b><i>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/>
<a NAME="E10:69_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as   <a href="finseq_2.html#NM2">FinSequence-DOMAIN</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> ;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:69_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> )
 ;<br/><b>thus </b><a NAME="E4:69_1_1_1"/>
<font color="Maroon">x</font> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E13:69_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> is  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 ;<br/>

<a NAME="E14:69_1_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E15:69_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/>


</div>
