<?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="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  not <font color="Olive" title="b1">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  &amp; <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  ) iff  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</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="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; (  for <font color="Olive" title="b3">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 <br/> (  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b3">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  iff ( <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> ) ) ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">V</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>;<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/>
<div><span class="lab"><font color="Green" title="E72">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:85_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:85_1"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:85_1"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<font color="Maroon" title="c4">x</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:85_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E3:85_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:85_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</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="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; (  for <font color="Olive" title="b3">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 <br/> (  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b3">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  iff ( <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> ) ) ) ) implies  ex <font color="Olive" title="b1">p</font> being   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  not <font color="Olive" title="b1">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  &amp; <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  ) )</span>
<div class="add"><span class="kw">given </span><font color="Maroon" title="c4">p</font> being   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E2:85_2"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
 not <font color="Maroon" title="c4">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> 
 <span class="kw">and </span><br/><a NAME="E3:85_2"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</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="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp; (  for <font color="Olive" title="b3">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 <br/> (  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b3">v</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  iff ( <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> ) ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c5">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:85_2"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T33" title="GRAPH_2:th.33">GRAPH_2:33</a></span>;<br/><a NAME="E6:85_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85_2"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="graph_2.html#D6" title="GRAPH_2:def.6">GRAPH_2:def 6</a></span>;<br/><a NAME="E7:85_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E8:85_2"/><span class="kw">then </span>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">vs</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">vs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">vs</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c7">v2</font> = <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">vs</font></span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c6">v1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v2</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="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v2</font> &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 <br/> (  <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>  iff ( <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v2</font> ) ) ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c9">v2</font> = <font color="Maroon" title="c7">v2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> &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 <br/> (  <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>  iff ( <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E10:85_2"/>
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_2"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E5:85_2"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="msscyc_1.html#D2" title="MSSCYC_1:def.2">MSSCYC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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 <br/> (  <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>  iff ( <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> ) )</span><br/><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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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>  iff ( <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> ) )</span><br/> <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c10">v</font> = 
 <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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th29</a></span>
<br/>.= 
 <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="c4">p</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_2"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:85_2"/>
(  <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>  iff ( <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">v2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_2"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E5:85_2"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="graph_3.html#T50" target="_self" title="GRAPH_3:th.50">Th55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</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><a NAME="E4:85"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font>
 <span class="kw">and </span><br/><a NAME="E5:85"/><span class="lab"><font color="Green" title="E74">A8</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 <br/> (  <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>  iff ( <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  not <font color="Olive" title="b1">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  &amp; <font color="Olive" title="b1">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">G9</font> =  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">E9</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>;<br/>
<a NAME="E6:85"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></span>;<br/>
<a NAME="E7:85"/>
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A10</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:85"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="#E6:85"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>;<br/>
<a NAME="E10:85"/>
 for <font color="Olive" title="b1">v</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> 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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">v9</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c8">v9</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></span>;<br/>
<a NAME="E2:85_3"/><span class="lab"><font color="Green" title="E78">A12</font></span>: 
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</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="c8">v9</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th29</a></span>;<br/>
<a NAME="E3:85_3"/><span class="lab"><font color="Green" title="E79">A13</font></span>: 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> (<font color="Maroon" title="c9">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>) <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="c9">v</font>, 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="#E9:85"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="graph_3.html#T31" target="_self" title="GRAPH_3:th.31">Th36</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:85_3_1"/>
( ( <font color="Maroon" title="c8">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c8">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> ) or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_1"/><span class="lab"><font color="Green" title="E80">A14</font></span>: 
( <font color="Maroon" title="c8">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c8">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><div class="add"><span class="kw">then </span> <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</font> = 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> (<font color="Maroon" title="c9">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_3"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="ref" href="graph_3.html#T48" target="_self" title="GRAPH_3:th.48">Th53</a></span>
<br/>.= 
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c9">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_3"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th29</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_1"/>
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</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="#E5:85"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E1:85_3_1_1"><span class="lab"><font color="Green" title="E80">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_2"/><span class="lab"><font color="Green" title="E80">A15</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">dv</font> =  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c9">v</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E74">A8</font></span></a></span>;<br/><a NAME="E3:85_3_1_2"/><span class="lab"><font color="Green" title="E81">A16</font></span>: 
<font color="Maroon" title="c10">dv</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ;<br/> <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</font> = 
<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>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p2">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E8:85"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="txt" href="#E2:85_3"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E1:85_3_1_2"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="graph_3.html#T47" target="_self" title="GRAPH_3:th.47">Th52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_3"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th29</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:85_3_1_2"/>
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c8">v9</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="#E3:85_3_1_2"><span class="lab"><font color="Green" title="E81">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">P9</font> being   <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  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)<span class="kw"> such that </span><br/><a NAME="E12:85"/><span class="lab"><font color="Green" title="E78">A17</font></span>: 
<font color="Maroon" title="c8">P9</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T59" target="_self" title="GRAPH_3:th.59">Th64</a></span>;<br/>
<a NAME="E13:85"/><span class="lab"><font color="Green" title="E79">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:85"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:85"/><span class="lab"><font color="Green" title="E80">A19</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P9</font>
 <span class="kw">and </span><br/><a NAME="E16:85"/><span class="lab"><font color="Green" title="E81">A20</font></span>: 
<font color="Maroon" title="c8">P9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E8:85"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">p9</font> being   <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  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)<span class="kw"> such that </span><br/><a NAME="E18:85"/><span class="lab"><font color="Green" title="E82">A21</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">and </span><br/><a NAME="E19:85"/><span class="lab"><font color="Green" title="E83">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font>
 <span class="kw">and </span><br/><a NAME="E20:85"/><span class="lab"><font color="Green" title="E84">A23</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p9</font> <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">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:85"><span class="lab"><font color="Green" title="E80">A19</font></span></a>, <a class="ref" href="graph_3.html#T10" target="_self" title="GRAPH_3:th.10">Th15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p</font> = (<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p9</font> as   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T5" target="_self" title="GRAPH_3:th.5">Th8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">vs9</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E23:85"/><span class="lab"><font color="Green" title="E85">A24</font></span>: 
<font color="Maroon" title="c12">vs9</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T33" title="GRAPH_2:th.33">GRAPH_2:33</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E86">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:85_4"/>
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</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="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:85_4"/><span class="lab"><font color="Green" title="E87">A26</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E4:85_4"/><span class="lab"><font color="Green" title="E88">A27</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:85_4"/><span class="lab"><font color="Green" title="E89">A28</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">and </span><br/><a NAME="E7:85_4"/><span class="lab"><font color="Green" title="E90">A29</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">and </span><br/><a NAME="E8:85_4"/>
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">and </span><br/><a NAME="E9:85_4"/><span class="lab"><font color="Green" title="E91">A30</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">and </span><br/><a NAME="E10:85_4"/>
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_4"><span class="lab"><font color="Green" title="E87">A26</font></span></a>, <a class="ref" href="graph_3.html#T2" target="_self" title="GRAPH_3:th.2">Th2</a></span>;<br/><a NAME="E11:85_4"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85_4"><span class="lab"><font color="Green" title="E89">A28</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>;<br/><a NAME="E12:85_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:85"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="txt" href="#E18:85"><span class="lab"><font color="Green" title="E82">A21</font></span></a>, <a class="txt" href="#E4:85_4"><span class="lab"><font color="Green" title="E88">A27</font></span></a>, <a class="txt" href="#E6:85_4"><span class="lab"><font color="Green" title="E89">A28</font></span></a>, <a class="txt" href="#E7:85_4"><span class="lab"><font color="Green" title="E90">A29</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E13:85_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:85_4"><span class="lab"><font color="Green" title="E91">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E25:85"/><span class="lab"><font color="Green" title="E87">A31</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:85"><span class="lab"><font color="Green" title="E80">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E26:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A32</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">p1</font> = (1,1) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p9</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="graph_2.html#T41" title="GRAPH_2:th.41">GRAPH_2:41</a></span>;<br/>
<a NAME="E28:85"/><span class="lab"><font color="Green" title="E89">A33</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c10">p9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="graph_2.html#T9" title="GRAPH_2:th.9">GRAPH_2:9</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">vs</font> = (<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font></span>)</span>) <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c12">vs9</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> ;<br/>
<a NAME="E30:85"/><span class="lab"><font color="Green" title="E90">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:85"><span class="lab"><font color="Green" title="E85">A24</font></span></a>, <a class="ref" href="graph_2.html#D6" title="GRAPH_2:def.6">GRAPH_2:def 6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_5"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c15">c</font> being    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c16">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:85_5"/><span class="lab"><font color="Green" title="E91">A35</font></span>: 
 not <font color="Maroon" title="c15">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E3:85_5"/>
<font color="Maroon" title="c16">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c15">c</font>
 <span class="kw">and </span><br/><a NAME="E4:85_5"/>
<font color="Maroon" title="c16">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 <span class="kw">and </span><br/><a NAME="E5:85_5"/>
<font color="Maroon" title="c16">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="ref" href="graph_3.html#T18" target="_self" title="GRAPH_3:th.18">Th23</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">c</font> = <font color="Maroon" title="c15">c</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></span>;<br/><a NAME="E7:85_5"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_5"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E8:85_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A36</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">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> &amp; <font color="Maroon" title="c17">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c17">c</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>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E9:85_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A37</font></span>: 
<font color="Maroon" title="c17">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>
 ;<br/><a NAME="E10:85_5"/>
<font color="Maroon" title="c17">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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 <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="txt" href="#E8:85_5"><span class="lab"><font color="Green" title="E92">A36</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:85_5"/><span class="lab"><font color="Green" title="E94">A38</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P9</font>
 <span class="kw">and </span><br/><a NAME="E13:85_5"/><span class="lab"><font color="Green" title="E95">A39</font></span>: 
<font color="Maroon" title="c8">P9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:85"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/><span class="kw">assume </span><a NAME="E14:85_5"/><span class="lab"><font color="Green" title="E96">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E15:85_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font> &amp; <font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:85_5"><span class="lab"><font color="Green" title="E94">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E16:85_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A41</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:85_5"><span class="lab"><font color="Green" title="E96">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E17:85_5"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="txt" href="#E14:85_5"><span class="lab"><font color="Green" title="E96">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E18:85_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:85"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="txt" href="#E9:85_5"><span class="lab"><font color="Green" title="E93">A37</font></span></a>, <a class="txt" href="#E13:85_5"><span class="lab"><font color="Green" title="E95">A39</font></span></a>, <a class="txt" href="#E16:85_5"><span class="lab"><font color="Green" title="E97">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:85"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E33:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A42</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E34:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A43</font></span>: 
<font color="Maroon" title="c14">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c11">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="txt" href="#E23:85"><span class="lab"><font color="Green" title="E85">A24</font></span></a>, <a class="txt" href="#E30:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="ref" href="graph_2.html#T42" title="GRAPH_2:th.42">GRAPH_2:42</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">vs</font> = <font color="Maroon" title="c14">vs</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c11">p</font> as   <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="txt" href="#E24:85"><span class="lab"><font color="Green" title="E86">A25</font></span></a>, <a class="ref" href="graph_3.html#T45" target="_self" title="GRAPH_3:th.45">Th50</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>  &amp; <font color="Maroon" title="c16">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>  )</span><br/>

<a NAME="E37:85"/><span class="lab"><font color="Green" title="E93">A44</font></span>: 
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T34" target="_self" title="GRAPH_3:th.34">Th39</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:85_6"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:85"><span class="lab"><font color="Green" title="E91">A42</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E2:85_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A45</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E3:85_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A46</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:85_6"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:85_6"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font>
 <span class="kw">and </span><br/><a NAME="E7:85_6"/><span class="lab"><font color="Green" title="E96">A47</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font>
 <span class="kw">and </span><br/><a NAME="E8:85_6"/><span class="lab"><font color="Green" title="E97">A48</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T1" title="GRAPH_2:th.1">GRAPH_2:1</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_6"><span class="lab"><font color="Green" title="E94">A45</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
;<br/><a NAME="E10:85_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A49</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:85_6"><span class="lab"><font color="Green" title="E97">A48</font></span></a></span>;<br/><a NAME="E11:85_6"/><span class="lab"><font color="Green" title="E99">A50</font></span>: 
<font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:85"><span class="lab"><font color="Green" title="E85">A24</font></span></a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></span>;<br/><a NAME="E12:85_6"/><span class="lab"><font color="Green" title="E100">A51</font></span>: 
( ( <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> ) or ( <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> &amp; <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:85"><span class="lab"><font color="Green" title="E93">A44</font></span></a>, <a class="txt" href="#E16:85"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="txt" href="#E26:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E18:85"><span class="lab"><font color="Green" title="E82">A21</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="txt" href="#E23:85"><span class="lab"><font color="Green" title="E85">A24</font></span></a>, <a class="txt" href="graph_3.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E13:85_6"/><span class="lab"><font color="Green" title="E101">A52</font></span>: 
<font color="Maroon" title="c15">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">vs</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E14:85_6"/>
<font color="Maroon" title="c15">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="txt" href="#E2:85_6"><span class="lab"><font color="Green" title="E94">A45</font></span></a>, <a class="txt" href="#E3:85_6"><span class="lab"><font color="Green" title="E95">A46</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:85_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E13:85_6"><span class="lab"><font color="Green" title="E101">A52</font></span></a>, <a class="txt" href="#E2:85_6"><span class="lab"><font color="Green" title="E94">A45</font></span></a>, <a class="txt" href="#E7:85_6"><span class="lab"><font color="Green" title="E96">A47</font></span></a>, <a class="txt" href="#E8:85_6"><span class="lab"><font color="Green" title="E97">A48</font></span></a>, <a class="txt" href="#E10:85_6"><span class="lab"><font color="Green" title="E98">A49</font></span></a>, <a class="txt" href="#E11:85_6"><span class="lab"><font color="Green" title="E99">A50</font></span></a>, <a class="txt" href="#E12:85_6"><span class="lab"><font color="Green" title="E100">A51</font></span></a>, <a class="txt" href="graph_3.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E39:85"/>
 not <font color="Maroon" title="c16">p</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:85"><span class="lab"><font color="Green" title="E92">A43</font></span></a>, <a class="txt" href="#E24:85"><span class="lab"><font color="Green" title="E86">A25</font></span></a>, <a class="ref" href="graph_3.html#T46" target="_self" title="GRAPH_3:th.46">Th51</a>, <a class="ref" href="msscyc_1.html#T6" title="MSSCYC_1:th.6">MSSCYC_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">p</font> is  <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a> </span><br/>

<a NAME="E40:85"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:85"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="txt" href="#E26:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E18:85"><span class="lab"><font color="Green" title="E82">A21</font></span></a>, <a class="txt" href="#E19:85"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="graph_2.html#T6" title="GRAPH_2:th.6">GRAPH_2:6</a></span>;<br/>
<a NAME="E41:85"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c16">p</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c16">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:85"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:85"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E6:85"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="txt" href="#E13:85"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E20:85"><span class="lab"><font color="Green" title="E84">A23</font></span></a>, <a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="xboole_1.html#T72" title="XBOOLE_1:th.72">XBOOLE_1:72</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">GRAPH_3:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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