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<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">pe</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">pe</font> &amp; not  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> );<br/>
<b>assume </b><a NAME="E1:28"/>
not  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c3">pe</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E26">A1</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>by </b><i><a class="ref" href="graph_5.html#T22" target="_self" title="GRAPH_5:th.22">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="graph_5.html#E2:28"><i><font color="Green" title="E26">A1</font></i></a>);<br/></i>
<a NAME="E5:28"/><i><font color="Green" title="E28">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">pe</font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E4:28"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:28"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:28"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:28"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:28"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E7:28"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="graph_5.html#E10:28"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E14:28"/><i><font color="Green" title="E31">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c3">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c12">r</font> = <font color="Maroon" title="c10">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E14:28"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">q</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">r</font>
;<br/>

<b>thus </b><a NAME="E16:28"/>
( <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">pe</font> &amp; not  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">pe</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E4:28"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="graph_5.html#E7:28"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><a NAME="E17:28"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c3">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">r</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E14:28"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q</font>
 ;<br/><a NAME="E2:28_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>assume </b><a NAME="E3:28_1"/><i><font color="Green" title="E34">A9</font></i>: 
not  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E4:28_1"/><i><font color="Green" title="E35">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">pe</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E14:28"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/><font color="Maroon" title="c11">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font> = 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graph_5.html#E1:28_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graph_5.html#E14:28"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="graph_5.html#E1:28_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">pe</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="graph_5.html#E1:28_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="graph_5.html#E4:28_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E6:28_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:28"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="graph_5.html#E1:28_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="graph_5.html#E3:28_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="graph_5.html#E4:28_1"><i><font color="Green" title="E35">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E7:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_5.html#E7:28"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="graph_5.html#E14:28"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="graph_5.html#E2:28_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:28"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T22" target="_self" title="GRAPH_5:th.22">Th22</a></i>;<br/>


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