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<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#V7" title="GRAPH_1:attr.7">finite</a> non <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E38">A1</font></i>: 
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c3">E</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">T</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font>;<br/>
 <a href="graph_1.html#K6" title="GRAPH_1:func.6">Degree</a> <font color="Maroon" title="c2">v</font> = 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c2">v</font>,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="graph_3.html#T29" target="_self" title="GRAPH_3:th.29">Th29</a></i>
<br/>.= 
<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="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</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="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>

;<br/>
<a NAME="E3:48"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:48"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
(  <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> <font color="Maroon" title="c2">v</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c6">e</font> <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="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">s</font> = 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> <font color="Maroon" title="c6">e</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E6:48"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_2"/>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1"><b>suppose </b></a><a NAME="E1:48_2_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E4:48"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="graph_3.html#E6:48"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_2"><b>suppose </b></a><a NAME="E1:48_2_2"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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="c9">vs</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="E3:48_2_2"/><i><font color="Green" title="E41">A4</font></i>: 
( not <font color="Maroon" title="c8">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c9">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="c2">v</font> &amp; <font color="Maroon" title="c9">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="c9">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T23" target="_self" title="GRAPH_3:th.23">Th23</a></i>;<br/><a NAME="E4:48_2_2"/>
<font color="Maroon" title="c8">c</font> is    <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="E5:48_2_2"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E6:48_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="graph_3.html#E3:48_2_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><a NAME="E7:48_2_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">c</font>
 ;<br/><a NAME="E8:48_2_2"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
<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="c8">c</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:48_2_2"/><b>then </b>
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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:48_2_2"/><b>then </b><i><font color="Green" title="E44">A7</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:48_2_2"/>
( <font color="Maroon" title="c9">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> 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> or <font color="Maroon" title="c9">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> 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> )
 <b>by </b><i><a class="txt" href="graph_3.html#E3:48_2_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="graph_3.html#E8:48_2_2"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="graph_3.html#E21"><i><font color="Green" title="E16">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E12:48_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E4:48"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="graph_3.html#E3:48_2_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="graph_3.html#E5:48_2_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="graph_3.html#E10:48_2_2"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a>, <a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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