<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a>   st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; ( for <font color="Olive" title="b2">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b2">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) holds <br/>( not  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Olive" title="b1">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c2">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) implies ( not  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">E</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">E9</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D9" title="GRAPH_1:def.9">GRAPH_1:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">ccp</font> = <font color="Maroon" title="c2">c</font> as   <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D8" target="_self" title="GRAPH_3:def.8">Def8</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:86"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E4:86"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
not <font color="Maroon" title="c2">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E5:86"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E6:86"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E7:86"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E8:86"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xx</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Olive" title="b1">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Erc</font> = <font color="Maroon" title="c4">E9</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v9</font> where <font color="Olive" title="b1">v9</font> is   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b1">v9</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="graph_3.html#T61" target="_self" title="GRAPH_3:th.61">Th61</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">c9</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K9" title="GRAPH_3:func.9">-CycleSet</a> , <font color="Maroon" title="c9">v</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E12:86"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E13:86"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
<font color="Maroon" title="c8">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Erc</font> <a href="graph_3.html#K13" title="GRAPH_3:func.13">-CycleSet</a> <font color="Maroon" title="c9">v</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:86"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
 <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="graph_3.html#D13" target="_self" title="GRAPH_3:def.13">Def13</a></span>;<br/>
<a NAME="E15:86"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:86"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>;<br/>
<a NAME="E16:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A8</font></span>: 
 ex <font color="Olive" title="b1">v9</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v9</font> &amp; <font color="Olive" title="b1">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> &amp;  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Olive" title="b1">v9</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> )
 
;<br/>
<div><span class="lab"><font color="Green" title="E77">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c6">Erc</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
(  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c10">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></span>;<br/><a NAME="E2:86_1"/>
(  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c4">E9</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> &amp;  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ccp</font></span>)</span> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th34</a>, <a class="ref" href="graph_3.html#T54" target="_self" title="GRAPH_3:th.54">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E3:86_1"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c6">Erc</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E78">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:86"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E19:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A11</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E20:86"/>
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c4">E9</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th34</a></span>;<br/>
<a NAME="E21:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A12</font></span>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c6">Erc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></span>;<br/>
<a NAME="E22:86"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:86"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E17:86"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:86"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E24:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A13</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E25:86"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:86"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E17:86"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E21:86"><span class="lab"><font color="Green" title="E79">A12</font></span></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></span>;<br/>
<a NAME="E26:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A14</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:86"/>
not  <a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E14:86"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="txt" href="#E24:86"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="ref" href="graph_3.html#T58" target="_self" title="GRAPH_3:th.58">Th58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">vr</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E29:86"/><span class="lab"><font color="Green" title="E82">A15</font></span>: 
<font color="Maroon" title="c10">vr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E30:86"/><span class="lab"><font color="Green" title="E83">A16</font></span>: 
 <a href="graph_3.html#K11" title="GRAPH_3:func.11">CatCycles</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c8">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c10">vr</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c10">vr</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:86"><span class="lab"><font color="Green" title="E81">A14</font></span></a>, <a class="txt" href="#E24:86"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="ref" href="graph_3.html#D10" target="_self" title="GRAPH_3:def.10">Def10</a></span>;<br/>
<a NAME="E31:86"/><span class="lab"><font color="Green" title="E84">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:86"><span class="lab"><font color="Green" title="E81">A14</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E32:86"/><span class="kw">then </span>
<font color="Maroon" title="c10">vr</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:86"><span class="lab"><font color="Green" title="E82">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E33:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c10">vr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="graph_3.html#E71"><span class="lab"><font color="Green" title="E59">Lm5</font></span></a></span>;<br/>
<a NAME="E34:86"/>
<font color="Maroon" title="c10">vr</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="graph_3.html#K1" title="GRAPH_3:func.1">-VSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:86"><span class="lab"><font color="Green" title="E84">A17</font></span></a>, <a class="txt" href="#E29:86"><span class="lab"><font color="Green" title="E82">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E35:86"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K10" title="GRAPH_3:func.10">Rotate</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c10">vr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="graph_3.html#E71"><span class="lab"><font color="Green" title="E59">Lm5</font></span></a></span>;<br/>
<a NAME="E36:86"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:86"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="txt" href="#E30:86"><span class="lab"><font color="Green" title="E83">A16</font></span></a>, <a class="txt" href="#E33:86"><span class="lab"><font color="Green" title="E85">A18</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E37:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A19</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c9</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:86"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></span>;<br/>
<a NAME="E38:86"/>
not <font color="Maroon" title="c8">c9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:86"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E17:86"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E21:86"><span class="lab"><font color="Green" title="E79">A12</font></span></a>, <a class="ref" href="graph_3.html#T60" target="_self" title="GRAPH_3:th.60">Th60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:86"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="graph_3.html#K14" title="GRAPH_3:func.14">ExtendCycle</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:86"><span class="lab"><font color="Green" title="E86">A19</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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