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

<b>set </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<a NAME="E1:76_1_1"/><i><font color="Green">E32</font></i>: 
not  <a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="graph_3.html#T7" target="_self">Th7</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_1_1"/>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> )
 </span><b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:76_1_1_1_1_1"/><b>then </b>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:76_1_1_1_1_1"/>
 ex <font color="Olive">e'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:76_1_1_1_1_2"/><b>then </b>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:76_1_1_1_1_2"/>
 ex <font color="Olive">e'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">e'</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e'</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">e'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:76_1_1"/><i><font color="Green">E36</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" target="_self">Edges_At</a> <font color="Maroon">v</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> &amp; <font color="Maroon">e'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">e'</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X'</font> ;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">e'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</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/>
<a NAME="E8:76_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:76_1_1"/>
 ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">t</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font>
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">s</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:76_1_1_2_1_1"/>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">s</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">s</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_2_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e'</font>
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:76_1_1_2_1_2"/>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="msscyc_1.html#T4">MSSCYC_1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:76_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:76_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:76_1_1_3"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 ;<br/><b>thus </b><a NAME="E4:76_1_1_3"/>
<font color="Maroon">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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:76_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">c</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> &amp; not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp;  ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span>  is   <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>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#D3">FINSEQ_2:def 3</a></i>;<br/>


</div>
