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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> ;<br/>





<b>assume </b><a NAME="E1:71"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">vv</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">vv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  ex <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; ( <font color="Maroon">vv</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> or <font color="Maroon">vv</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> ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; ( <font color="Maroon">vv</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> or <font color="Maroon">vv</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> ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Cset</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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E8:71"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36">GRAPH_2:36</a></i>;<br/>
<a NAME="E9:71"/>
<font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T3" target="_self">Th3</a>, <a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_2.html#T34">GRAPH_2:34</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:71"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:71"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E13:71"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:71"/><b>then </b><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i><a class="ref" href="graph_3.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1"/>
( ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_1"/>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><div class="add"><a NAME="E2:71_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/><a NAME="E3:71_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:71_1_1_1"/>
not <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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="graph_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_2"/><b>then </b>
<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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_3.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E3:71_1_1_2"/>
not <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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_3"><b>suppose </b></a><a NAME="E1:71_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_3"/><b>then </b>
<font color="Maroon">vs</font> <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="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_3.html#T7" target="_self">Th7</a>, <a class="ref" href="msscyc_1.html#T6">MSSCYC_1:6</a></i>;<br/><a NAME="E3:71_1_1_3"/><b>then </b>
<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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a></i>;<br/><b>hence </b><a NAME="E4:71_1_1_3"/>
not <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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_4"><b>suppose </b></a><a NAME="E1:71_1_1_4"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#T1">FSM_1:1</a></i>;<br/><a NAME="E4:71_1_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self">Th8</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:71_1_1_4"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:71_1_1_4"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="graph_3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = 1,<font color="Maroon">m</font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c'</font>, <font color="Maroon">c2</font> = <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c'</font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">vs1</font> = 1,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E8:71_1_1_4"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:71_1_1_4"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">1,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E10:71_1_1_4"/><i><font color="Green">E81</font></i>: 
1,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, , , <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><b>set </b><font color="Maroon">vs2</font> = <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E11:71_1_1_4"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E12:71_1_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="graph_3.html#T8" target="_self">Th8</a>, , <a class="ref" href="graph_3.html#T9" target="_self">Th9</a>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><b>reconsider </b><font color="Maroon">vs21</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K4">^'</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_4_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:71_1_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:71_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E15:71_1_1_4"/><b>then </b><i><font color="Green">E90</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span>
 ;<br/><a NAME="E16:71_1_1_4"/><b>then </b><i><font color="Green">E91</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E93</font></i>: <font color="Maroon">vs21</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>
<br/>.= 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i>, <a class="txt" href="graph_3.html#E4:71_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
;<br/><b>consider </b><font color="Maroon">lv2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:71_1_1_4"/><i><font color="Green">E94</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lv2</font> &amp; <font color="Maroon">lv2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lv2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_3.html#E6:71_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><a NAME="E20:71_1_1_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lv2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E8:71_1_1_4"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E21:71_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E15:71"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E22:71_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span>
 ;<br/><a NAME="E23:71_1_1_4"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E125</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lv2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="graph_3.html#E8:71_1_1_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="graph_3.html#E5"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="graph_3.html#E9:71_1_1_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_3.html#E10:71_1_1_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="msscyc_1.html#T6">MSSCYC_1:6</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:71_1_1_4_4"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font> )
 ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_4_4"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:71_1_1_4_4"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:71_1_1_4_4"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T13" target="_self">Th13</a>, </i>;<br/><b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:71_1_1_4_4"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:71_1_1_4_4"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E11:71_1_1_4_4"/>
contradiction
 <b>by </b><i><a class="ref" href="graph_3.html#T13" target="_self">Th13</a>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E26:71_1_1_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c21'</font> = <font color="Maroon">c2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c1</font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_3.html#E11:71_1_1_4"><i><font color="Green">E82</font></i></a>, </i>;<br/><a NAME="E28:71_1_1_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">vs21</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c21'</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_3.html#E11:71_1_1_4"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_2.html#T47">GRAPH_2:47</a></i>;<br/><a NAME="E29:71_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E15:71"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E30:71_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">vs21</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">vs21</font></span>)</span> = 
<span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">1,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>
<br/>.= 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c21'</font> = <font color="Maroon">c21'</font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="graph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c21</font> = <font color="Maroon">c21'</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  <b>by </b><i/>;<br/><a NAME="E34:71_1_1_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T9">GRAPH_2:9</a></i>;<br/> <a href="relat_1.html#K2">rng</a> <font color="Maroon">c21</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="graph_3.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E36:71_1_1_4"/><b>then </b>
<font color="Maroon">c21</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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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><a class="ref" href="graph_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="graph_3.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E37:71_1_1_4"/>
not <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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E16:71"/>
<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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_2"/>
<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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> <b> such that </b><br/><a NAME="E3:71_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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="E4:71_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:71"/>
<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">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">c'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &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  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div>
