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

<b>set </b><font color="Maroon" title="c4">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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">c</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="c2">v</font> ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span>;<br/>
<a NAME="E1:76_1_1"/><i><font color="Green" title="E64">A2</font></i>: 
not  <a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:76"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="graph_3.html#T30" target="_self" title="GRAPH_3:th.30">Th30</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" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E1:76_1_1"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )</span><br/><div class="add"><a NAME="E2:76_1_1_1_1_1"/><b>then </b>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )</span><br/><div class="add"><a NAME="E2:76_1_1_1_1_2"/><b>then </b>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">e'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e'</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e'</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">e'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E4:76_1_1"/><i><font color="Green" title="E65">A3</font></i>: 
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="graph_3.html#K4" title="GRAPH_3:func.4">Edges_At</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e'</font> &amp; <font color="Maroon" title="c6">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">e'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e'</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X'</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="graph_3.html#E4:76_1_1"><i><font color="Green" title="E65">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">e'</font> = <font color="Maroon" title="c6">e'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X'</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">c</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">e'</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:76_1_1"/><i><font color="Green" title="E66">A4</font></i>: 
<font color="Maroon" title="c9">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E4:76_1_1"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="graph_3.html#T7" target="_self" title="GRAPH_3:th.7">Th7</a></i>;<br/>
<a NAME="E9:76_1_1"/>
 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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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="c9">c</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="c2">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" title="c10">t</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">t</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E4:76_1_1"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">s</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">s</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E4:76_1_1"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">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" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font> )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E4:76_1_1"><i><font color="Green" title="E65">A3</font></i></a></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" title="E67">A5</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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="c9">c</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="c2">v</font> )</span><br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c13">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c13">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c13">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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="c2">v</font> )</span><br/><b>thus </b><a NAME="E2:76_1_1_2_1_1"/>
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c13">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c13">s</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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="c2">v</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:76_1_1_2_1_1"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="graph_3.html#T16" target="_self" title="GRAPH_3:th.16">Th16</a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E67">A6</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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="c9">c</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="c2">v</font> )</span><br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c11">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c11">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c11">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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="c2">v</font> )</span><br/><b>thus </b><a NAME="E2:76_1_1_2_1_2"/>
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c11">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">c</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c11">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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="c2">v</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:76_1_1_2_1_2"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a>, <a class="ref" href="msscyc_1.html#T4" title="MSSCYC_1:th.4">MSSCYC_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E67">A7</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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">c</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="c2">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E8:76_1_1"><i><font color="Green" title="E66">A4</font></i></a></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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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">c</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="c2">v</font> ) ) </span>}</span>  implies <font color="Maroon" title="c10">x</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> )</span><br/><b>assume </b><a NAME="E1:76_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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">c</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="c2">v</font> ) ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font></span><br/><b>then consider </b><font color="Maroon" title="c11">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E3:76_1_1_3"/><i><font color="Green" title="E68">A8</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Maroon" title="c11">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c11">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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="c11">c</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="c2">v</font> ) )
 ;<br/><b>thus </b><a NAME="E4:76_1_1_3"/>
<font color="Maroon" title="c10">x</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:76_1_1_3"><i><font color="Green" title="E68">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:76_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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">c</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="c2">v</font> ) ) </span>}</span>  is   <a href="finseq_2.html#NM2" title="FINSEQ_2:NM.2">FinSequence-DOMAIN</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E10:76_1_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="finseq_2.html#D3" title="FINSEQ_2:def.3">FINSEQ_2:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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