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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">pe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">FS</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">FT</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E2:34"/><i><font color="Green">E45</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">pe</font>
 ;<br/><a NAME="E2:34_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:34_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:34_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E7:34_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:34_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:34_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 ;<br/><a NAME="E3:34_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E4:34_1"/><b>then </b>
( <font color="Maroon">x</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> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> or <font color="Maroon">x</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> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:34_1"/><b>then </b>
( <font color="Maroon">x</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> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> or <font color="Maroon">x</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> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/><a NAME="E7:34_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">pe</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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