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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">connected</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">vs</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green" title="E72">A1</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="graph_3.html#V1" title="GRAPH_3:attr.1">Eulerian</a>
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c3">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green" title="E74">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">vs</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E4:84"/><i><font color="Green" title="E75">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">vs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:84"/><i><font color="Green" title="E76">A5</font></i>: 
( ( <font color="Maroon" title="c4">x</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="c3">vs</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font> &amp; not <font color="Maroon" title="c4">x</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="c3">vs</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:84"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E7:84"/>
<font color="Maroon" title="c3">vs</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:84"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="graph_3.html#T3" target="_self" title="GRAPH_3:th.3">Th3</a></i>;<br/>
<a NAME="E8:84"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">vs</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:84"/><i><font color="Green" title="E77">A6</font></i>: 
<font color="Maroon" title="c5">y</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="c3">vs</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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="c7">vs1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E12:84"/><i><font color="Green" title="E78">A7</font></i>: 
not <font color="Maroon" title="c6">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E13:84"/><i><font color="Green" title="E79">A8</font></i>: 
<font color="Maroon" title="c7">vs1</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c6">c</font>
 <b>and </b><br/><a NAME="E14:84"/><i><font color="Green" title="E80">A9</font></i>: 
( <font color="Maroon" title="c7">vs1</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">x</font> &amp; <font color="Maroon" title="c7">vs1</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="c7">vs1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E4:84"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="graph_3.html#E6:84"><i><font color="Green" title="E76">A5</font></i></a>, <a class="txt" href="graph_3.html#E10:84"><i><font color="Green" title="E77">A6</font></i></a>, <a class="ref" href="graph_3.html#T23" target="_self" title="GRAPH_3:th.23">Th23</a></i>;<br/>
<a NAME="E15:84"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="graph_3.html#E12:84"><i><font color="Green" title="E78">A7</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E16:84"/><b>then </b><i><font color="Green" title="E81">A10</font></i>: 
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="c6">c</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c6">c</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></i>;<br/>
<a NAME="E18:84"/>
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="c8">c</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E16:84"><i><font color="Green" title="E81">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:84"/><b>then </b><i><font color="Green" title="E82">A11</font></i>: 
<font color="Maroon" title="c8">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="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E20:84"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:84"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="graph_3.html#D14" target="_self" title="GRAPH_3:def.14">Def14</a></i>;<br/>
<a NAME="E21:84"/><b>then </b>
( the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <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="c3">vs</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <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="c3">vs</font> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:84"><i><font color="Green" title="E73">A2</font></i></a>, <a class="txt" href="graph_3.html#E19:84"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="graph_3.html#T20" target="_self" title="GRAPH_3:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E22:84"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E4:84"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="graph_3.html#E6:84"><i><font color="Green" title="E76">A5</font></i></a>, <a class="txt" href="graph_3.html#E13:84"><i><font color="Green" title="E79">A8</font></i></a>, <a class="txt" href="graph_3.html#E14:84"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="graph_3.html#E16:84"><i><font color="Green" title="E81">A10</font></i></a>, <a class="txt" href="graph_3.html#E21"><i><font color="Green" title="E16">Lm3</font></i></a></i>;<br/>


</div>
