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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">c1</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> , <font color="Maroon">c2</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><b>that </b><br/><a NAME="E1:75"/><i><font color="Green">E30</font></i>: 
<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">c1</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</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">c2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green">E32</font></i>: 
 <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>and </b><br/><a NAME="E3:75"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:75"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:75"/><i><font color="Green">E46</font></i>: 
 <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c1</font>,<font color="Maroon">v</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c2</font>,<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#T8" target="_self">Th8</a>, <a class="ref" href="graph_3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:75"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="graph_3.html#K1" target="_self">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_3.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:75"/><b>then </b><i><font color="Green">E47</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">c1</font></span>)</span> &amp; <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">c2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1"/>
( <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><b>suppose </b></a><a NAME="E1:75_1_1"/>
<font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:75_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E3:75_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c1</font>,<font color="Maroon">v</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:75_1_1"/>
not  <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</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="75_1_2"><b>suppose </b></a><a NAME="E1:75_1_2"/>
<font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:75_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E3:75_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" target="_self">Rotate</a> <font color="Maroon">c2</font>,<font color="Maroon">v</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:75_1_2"/>
not  <a href="graph_3.html#K11" target="_self">CatCycles</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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