<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon">ee</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 3</span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:48"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">ee</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> ) </span>}</span>  &amp;  <a href="sgraph1.html#K8" target="_self">TriangleGraph</a>  <a href="hidden.html#R1">=</a>  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 3</span>)</span>,<font color="Maroon">ee</font> #) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">ee</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ee</font>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E3:48_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_1_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, <a class="txt" href="sgraph1.html#E1:48_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, <a class="txt" href="sgraph1.html#E1:48_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, <a class="txt" href="sgraph1.html#E1:48_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_3_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_3_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_3_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_3_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_3_1_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_3_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">ee</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_2_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">3,1</span><a href="tarski.html#K2">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1"><b>suppose </b></a><a NAME="E1:48_2_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = 1;<br/><b>take </b><font color="Maroon">j</font> = 2;<br/><b>thus </b><a NAME="E1:48_2_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>thus </b><a NAME="E2:48_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/><b>thus </b><a NAME="E3:48_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_2_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ee</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_2"><b>suppose </b></a><a NAME="E1:48_2_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = 2;<br/><b>take </b><font color="Maroon">j</font> = 3;<br/><b>thus </b><a NAME="E1:48_2_1_2_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_2_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ee</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_3"><b>suppose </b></a><a NAME="E1:48_2_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">3,1</span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = 3;<br/><b>take </b><font color="Maroon">j</font> = 1;<br/><b>thus </b><a NAME="E1:48_2_1_3_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 3 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_2_1_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ee</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:48"/><b>then </b>
<span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">ee</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
( <font color="Maroon">ee</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">1,2</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">2,3</span><a href="setwiseo.html#K3">.}</a></span>,<span class="p2"><a href="setwiseo.html#K3">{.</a><span class="default">3,1</span><a href="setwiseo.html#K3">.}</a></span></span><a href="setwiseo.html#K4">.}</a></span> &amp;  <a href="sgraph1.html#K8" target="_self">TriangleGraph</a>  <a href="hidden.html#R1">=</a>  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 3</span>)</span>,<font color="Maroon">ee</font> #) )
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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