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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_3.html#NM2">Vertex</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_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> ;<br/>





<b>assume </b><a NAME="E1:71"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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<sub>3</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>4</sub></font> <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="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <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="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>4</sub></font> <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>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <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>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#D8" target="_self">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <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><b> such that </b><br/><a NAME="E8:71"/><i><font color="Green">E62</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_2.html#T36">GRAPH_2:36</a></i>;<br/>
<a NAME="E9:71"/>
<font color="Maroon">c<sub>1</sub></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<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E5:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</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">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:71"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:71"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</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>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</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">c<sub>8</sub></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">c<sub>8</sub></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">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E11:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green">E66</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">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> ) or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <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> ) )
 </span><b>by </b><i><a class="txt" href="graph_3.html#E14:71"><i><font color="Green">E65</font></i></a>, <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">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></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<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</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<sub>3</sub></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">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E5:71"><i><font color="Green">E61</font></i></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">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E11:71"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E3:71_1_1_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:71_1_1_3"/><b>then </b>
<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>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E11:71"><i><font color="Green">E63</font></i></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<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</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">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">E67</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <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> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_4"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></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">E69</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <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="txt" href="graph_3.html#E12:71"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></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">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></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">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <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="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E4:71_1_1_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = 1,<font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>,<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="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_3.html#T8" target="_self">Th8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E8:71_1_1_4"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <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">E72</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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E10:71_1_1_4"/><i><font color="Green">E73</font></i>: 
1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E6:71_1_1_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font>,<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="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E11:71_1_1_4"/><i><font color="Green">E74</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">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</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>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <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">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<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="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E12:71"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E4:71_1_1_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></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">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E11:71_1_1_4"><i><font color="Green">E74</font></i></a></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">E76</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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E16:71_1_1_4"/><b>then </b><i><font color="Green">E77</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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">E78</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E15:71_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">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:71_1_1_4"/><i><font color="Green">E79</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></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">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_3.html#E16:71_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">E80</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E11:71_1_1_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_3.html#E19:71_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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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#E9:71_1_1_4"><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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E23:71_1_1_4"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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><a class="txt" href="graph_3.html#E8:71_1_1_4"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><i><font color="Green">E82</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E19:71_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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="graph_3.html#E8:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_3.html#E20:71_1_1_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_3.html#E23:71_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">c<sub>17</sub></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">E83</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_4_4"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:71_1_1_4_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:71_1_1_4_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></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">c<sub>18</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71_1_1_4_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:71_1_1_4_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <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="txt" href="graph_3.html#E5:71_1_1_4_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E10:71_1_1_4_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E7:71_1_1_4_4"><i><font color="Green">E86</font></i></a>, <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="txt" href="graph_3.html#E3:71_1_1_4_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_3.html#E5:71_1_1_4_4"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_3.html#E7:71_1_1_4_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_3.html#E9:71_1_1_4_4"><i><font color="Green">E87</font></i></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">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></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">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E10:71_1_1_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E12:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_3.html#E24:71_1_1_4"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_3.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E28:71_1_1_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E10:71_1_1_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_3.html#E12:71_1_1_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_3.html#E24:71_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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E9:71_1_1_4"><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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">c<sub>15</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>15</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="graph_3.html#E1:71_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_3.html#E17:71_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_3.html#E28:71_1_1_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="msscyc_1.html#D2">MSSCYC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  <b>by </b><i><a class="ref" href="graph_3.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E34:71_1_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_3.html#E6:71_1_1_4"><i><font color="Green">E70</font></i></a>, <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">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></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<sub>3</sub></font>
<b>by </b><i><a class="txt" href="graph_3.html#E34:71_1_1_4"><i><font color="Green">E84</font></i></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_3.html#E11:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_3.html#E17:71_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_3.html#E28:71_1_1_4"><i><font color="Green">E83</font></i></a></i>;<br/><b>hence </b><a NAME="E37:71_1_1_4"/>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></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">E67</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:71_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E3:71_2"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:71"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a>  : (  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="graph_3.html#K9" target="_self">-CycleSet</a> </span>)</span>
 <b>by </b><i><a class="txt" href="graph_3.html#E15:71"><i><font color="Green">E66</font></i></a></i>;<br/>


</div>
