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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>set </b><font color="Maroon" title="c4">vsc1</font> =  <a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">vsc2</font> =  <a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">c2</font>;<br/>
<a NAME="E1:25"/><i><font color="Green" title="E15">A1</font></i>: 
(  <a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">c1</font> &amp;  <a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">c2</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c3">c2</font> )
 
<b>by </b><i><a class="ref" href="graph_2.html#D11" title="GRAPH_2:def.11">GRAPH_2:def 11</a></i>;<br/>
<a NAME="E2:25"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font></span>)</span> <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">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="graph_2.html#D7" title="GRAPH_2:def.7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">c2</font></span>)</span> <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">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_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">c2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">c1c2</font> = <font color="Maroon" title="c2">c1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">c2</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="msscyc_1.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="msscyc_1.html#E2:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="graph_2.html#T46" title="GRAPH_2:th.46">GRAPH_2:46</a></i>;<br/><a NAME="E3:25_1"/>
<font color="Maroon" title="c6">c1c2</font> is <a href="graph_1.html#V8" title="GRAPH_1:attr.8">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a><br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c7">n</font> <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="c6">c1c2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
( <font color="Maroon" title="c7">n</font> <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="c2">c1</font> or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font> or <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <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="c2">c1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c2">c1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:25_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">c1</font> &amp; <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">c1</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="msscyc_1.html#E1:25_1_1_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:25_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_1_1"/>
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"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="msscyc_1.html#E1:25_1_1_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_2"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_2"/><i><font color="Green" title="E21">A8</font></i>: 
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="msscyc_1.html#T14" target="_self" title="MSSCYC_1:th.14">Th14</a></i>;<br/><a NAME="E3:25_1_1_1_2"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">c1</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:25_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_1_2"/>
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"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="msscyc_1.html#E1:25_1_1_1_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="msscyc_1.html#E2:25_1_1_1_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="graph_2.html#D11" title="GRAPH_2:def.11">GRAPH_2:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_3"><b>suppose </b></a><a NAME="E1:25_1_1_1_3"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">c1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E3:25_1_1_1_3"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:25_1_1_1_3"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:25_1_1_1_3"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E5:25_1_1_1_3"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c8">k</font> <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="c3">c2</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="msscyc_1.html#E1:25_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E6:25_1_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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">c2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:25_1_1_1_3"/><b>then </b><i><font color="Green" title="E23">A12</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">c2</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E4:25_1_1_1_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="msscyc_1.html#E5:25_1_1_1_3"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:25_1_1_1_3"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c7">n</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">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">k</font>
 ;<br/><a NAME="E9:25_1_1_1_3"/>
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><a NAME="E10:25_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E7:25_1_1_1_3"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="msscyc_1.html#E8:25_1_1_1_3"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E11:25_1_1_1_3"/>
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"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E4:25_1_1_1_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="msscyc_1.html#E5:25_1_1_1_3"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon" title="c2">c1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">c2</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:25"/>
<font color="Maroon" title="c2">c1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">c2</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">c1c2</font> = <font color="Maroon" title="c2">c1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">c2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>set </b><font color="Maroon" title="c7">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font>;<br/>
<a NAME="E7:25"/><i><font color="Green" title="E18">A14</font></i>: 
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#T14" target="_self" title="MSSCYC_1:th.14">Th14</a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green" title="E19">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">c1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green" title="E20">A16</font></i>: 
( 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="c2">c1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font> <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="c2">c1</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:25"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">c2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
0 <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="c3">c2</font>
 
;<br/>
<a NAME="E12:25"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font> <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="c6">c1c2</font>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E3:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E13:25"/><b>then </b><i><font color="Green" title="E21">A17</font></i>: 
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"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E9:25"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></i>;<br/>
<a NAME="E14:25"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">c2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E15:25"/><b>then </b>
( <font color="Maroon" title="c6">c1c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c6">c1c2</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="c2">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</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="c2">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E8:25"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E16:25"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="msscyc_1.html#E7:25"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="msscyc_1.html#E13:25"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="graph_2.html#D11" title="GRAPH_2:def.11">GRAPH_2:def 11</a></i>;<br/>


</div>
