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<b>let </b><font color="Maroon">G</font> be  <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be  <a href="msscyc_1.html#V1">cyclic</a> <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/>
<font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">rc</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E3:67_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">rc</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">rc</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/><a NAME="E4:67_1_1"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<font color="Maroon">rc</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0
 ;<br/><b>hence </b><a NAME="E5:67_1_1"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> is <a href="abian.html#V1">even</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:67_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36">GRAPH_2:36</a></i>;<br/><b>thus </b><a NAME="E4:67_1_2"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> is <a href="abian.html#V1">even</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_1_2_1_1_1"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2"/><b>then </b>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:67_1_2_1_1_2"/>
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> is <a href="abian.html#V1">even</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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