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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>6</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/>
<b>assume </b><a NAME="E4:19"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E8:19"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> is    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E4:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E6:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="graph_2.html#T46">GRAPH_2:46</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E4:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E6:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="graph_3.html#E7:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="graph_2.html#T46">GRAPH_2:46</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E10:19"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="graph_2.html#K3">^'</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/>

<a NAME="E11:19"/>
<font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="graph_3.html#E4:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graph_3.html#E6:19"><i><font color="Green">E18</font></i></a>, <a class="txt" href="graph_3.html#E7:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/>
<b>hence </b><a NAME="E12:19"/>
<font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E13:19"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E14:19"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E15:19"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/>

<a NAME="E16:19"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E17:19"/><b>then </b>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E18:19"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>;<br/>


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