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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:23_7_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="glib_000.html#V4" target="_self">trivial</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a> )
 ;<br/>

<a NAME="E2:23_7_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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="glib_000.html#D21" target="_self">Def21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_7_1"/><i><font color="Green">E20</font></i>: 
 <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></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_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_7_1_1"/><i><font color="Green">E21</font></i>: 
not  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><a NAME="E2:23_7_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="glib_000.html#E1:23_7_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E3:23_7_1_1"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:23_7_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:23_7_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:23_7_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:23_7_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:23_7_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="glib_000.html#D16" target="_self">Def16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_7_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:23_7_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:23_7_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:23_7_1_1"><i><font color="Green">E22</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_7_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E3:23_7_1_1_1"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E2:23_7_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:23_7_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:23_7_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:23_7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:23_7_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="glib_000.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E6:23_7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:23_7_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="glib_000.html#E5:23_7_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="glib_000.html#D23" target="_self">Def23</a></i>;<br/><b>hence </b><a NAME="E7:23_7_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:23_7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_000.html#E1:23_7_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:23_7_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="glib_000.html#V2" target="_self">finite</a>
 <b>by </b><i><a class="txt" href="glib_000.html#E4:23_7_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_000.html#D19" target="_self">Def19</a></i>;<br/>


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