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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:23_6_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">G</font> is <a href="glib_000.html#V4" target="_self">trivial</a> &amp; <font color="Maroon">G</font> is <a href="glib_000.html#V3" target="_self">loopless</a> )
 ;<br/>

<a NAME="E2:23_6_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_6_1"/><i><font color="Green">E30</font></i>: 
 <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_6_1_1_1"/>
(  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</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="23_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_6_1_1_1_1"/>
 <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_6_1_1_1_1"/>
 <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_6_1_1_1_2"/>
not  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_6_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:23_6_1_1_1_2"/>
( <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#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G</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#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:23_6_1_1_1_2"/><b>then </b>
( <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="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:23_6_1_1_1_2"/>
 <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:23_6_1"/>
<font color="Maroon">G</font> is <a href="glib_000.html#V2" target="_self">finite</a>
 <b>by </b><i>, </i>;<br/>


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