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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>
<div><i><font color="Green">E4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K1" target="_self">-SVSet</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K1" target="_self">-SVSet</a> <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E3:8_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K1" target="_self">-SVSet</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_4.html#E1:8_1"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E4:8_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:8_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:8_1"><i><font color="Green">E6</font></i></a></i>;<br/><b>thus </b><a NAME="E9:8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_4.html#E8:8_1"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K2" target="_self">-TVSet</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K2" target="_self">-TVSet</a> <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E3:8_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K2" target="_self">-TVSet</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_4.html#E1:8_2"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E4:8_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:8_2"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:8_2"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:8_2"><i><font color="Green">E6</font></i></a></i>;<br/><b>thus </b><a NAME="E9:8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_4.html#E8:8_2"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K1" target="_self">-SVSet</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#K2" target="_self">-TVSet</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:8"><i><font color="Green">E4</font></i></a></i>;<br/>


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