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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2" target="_self">finite</a> <a href="glib_000.html#V3" target="_self">loopless</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">In</font> = <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> ;<br/>
<b>set </b><font color="Maroon">Out</font> = <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:120_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 ;<br/><a NAME="E2:120_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D19" target="_self">Def19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:120_1"/><b>then </b>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E14"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:120_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:120"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:120"/><b>then </b>
<font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:120"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<b>hence </b><a NAME="E5:120"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a> </span>)</span>
 ;<br/>


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