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<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>set </b><font color="Maroon">T</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>assume </b><a NAME="E1:41"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> ) )
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E32</font></i>: 
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:41"/>
 <a href="graph_3.html#K5" target="_self">Edges_In</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:41"/>
 <a href="graph_3.html#K6" target="_self">Edges_Out</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:41"/><i><font color="Green">E46</font></i>: 
 <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">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E8:41"/><i><font color="Green">E47</font></i>: 
 <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">X</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:41_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
 <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="txt" href="graph_3.html#E6:41"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:41_1_1"/><b>then </b>
 <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">X</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:41"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_3.html#E4:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/><div class="add"><a NAME="E2:41_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:41_1_2"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_1_2"/><b>then </b>
 <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<font color="Maroon">X</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E6:41"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:41_1_2"/><b>then </b>
 <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">X</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E6:41_1_2"/>
 <a href="graph_1.html#K6">Degree</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2:41"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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