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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">vs</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c4">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="c3">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">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="c4">e</font>, <font color="Maroon" title="c6">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="c4">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="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">ec</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="msscyc_1.html#T5" title="MSSCYC_1:th.5">MSSCYC_1:5</a></i>;<br/>
<b>assume </b><a NAME="E4:18"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">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="c3">vs</font></span>)</span> <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="c4">e</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">vse</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">t</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <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> ;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c8">vse</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c7">ec</font>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#T4" title="MSSCYC_1:th.4">MSSCYC_1:4</a></i>;<br/>
<a NAME="E7:18"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c8">vse</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
<font color="Maroon" title="c2">c</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="graph_3.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="graph_3.html#E6:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="graph_2.html#T46" title="GRAPH_2:th.46">GRAPH_2:46</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">ce</font> = <font color="Maroon" title="c2">c</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">ec</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="graph_3.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="graph_3.html#E6:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="graph_3.html#E7:18"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="graph_2.html#T46" title="GRAPH_2:th.46">GRAPH_2:46</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">vs'</font> = <font color="Maroon" title="c3">vs</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c8">vse</font>;<br/>

<b>thus </b><a NAME="E10:18"/>
<font color="Maroon" title="c10">vs'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">vs</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default">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="c4">e</font></span>)</span>,<span class="p2">(<span class="default">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="c4">e</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ;<br/>

<a NAME="E11:18"/>
<font color="Maroon" title="c10">vs'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">ce</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="graph_3.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="graph_3.html#E6:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="graph_3.html#E7:18"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="graph_2.html#T47" title="GRAPH_2:th.47">GRAPH_2:47</a></i>;<br/>
<b>hence </b><a NAME="E12:18"/>
<font color="Maroon" title="c10">vs'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ;<br/>

<a NAME="E13:18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c2">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E14:18"/><b>then </b>
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">vs</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E15:18"/>
<font color="Maroon" title="c10">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="c3">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/>

<a NAME="E16:18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">vse</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E17:18"/><b>then </b>
( 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="c8">vse</font> &amp; <font color="Maroon" title="c8">vse</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">vse</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E18:18"/>
<font color="Maroon" title="c10">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="c10">vs'</font></span>)</span> <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="c4">e</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></i>;<br/>


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