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

<b>set </b><font color="Maroon" title="c1">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E1:49"/><i><font color="Green" title="E27">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">VERTICES</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">One</font> = 1, <font color="Maroon" title="c4">Two</font> = 2, <font color="Maroon" title="c5">Three</font> = 3 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">VERTICES</font> <b>by </b><i><a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">o</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a>  <b>by </b><i><a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">ONE</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">One</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>, <font color="Maroon" title="c8">TWO</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">Two</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>, <font color="Maroon" title="c9">THREE</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">Three</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a>  ;<br/>
<a NAME="E6:49"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">ONE</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">TWO</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">THREE</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">ONE</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">ONE</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">TWO</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">THREE</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">pp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><i><font color="Green" title="E28">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</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="E1:49_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> )
 ;<br/><a NAME="E2:49_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/><a NAME="E3:49_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:49_1"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="sgraph1.html#E4:49_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a> 
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a> 
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_3"><b>suppose </b></a><a NAME="E1:49_1_1_3"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_3"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a> 
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="sgraph1.html#T46" target="_self" title="SGRAPH1:th.46">Th46</a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</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="E1:49_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">pp</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">pp</font> )
 ;<br/><a NAME="E2:49_2"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:49_2"/><i><font color="Green" title="E31">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 <b>by </b><i><a class="txt" href="sgraph1.html#E2:49_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:49_2"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K1" title="SGRAPH1:func.1">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,3 )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:49_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="sgraph1.html#E5:49_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:49_2_1_1"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/><a NAME="E3:49_2_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or ( 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 ) )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:49_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:49_2_1_1"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or ( 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 ) )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_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="E1:49_2_1_1_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="sgraph1.html#E4:49_2_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_1"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_1"/>
<font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E2:49_2_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/><b>thus </b><a NAME="E3:49_2_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_1_1_1_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_2"/>
<font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E2:49_2_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="finseq_1.html#T87" title="FINSEQ_1:th.87">FINSEQ_1:87</a></i>;<br/><b>thus </b><a NAME="E3:49_2_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_1_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:49_2_1_1"/>
( <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_2"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:49_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:49_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_2"/>
( <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="sgraph1.html#E1:49_2_1_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T10" title="ZFMISC_1:th.10">ZFMISC_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_3"><b>suppose </b></a><a NAME="E1:49_2_1_3"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_3"/>
( <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E2:49_2"><i><font color="Green" title="E29">A4</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></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:49"/><b>then </b>
<font color="Maroon" title="c10">pp</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c6">o</font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E1:49"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="sgraph1.html#E8:49"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="sgraph1.html#D12" target="_self" title="SGRAPH1:def.12">Def12</a></i>;<br/>
<a NAME="E11:49"/><b>then </b>
<font color="Maroon" title="c10">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K5" title="SGRAPH1:func.5">PATHS</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c6">o</font>
 
;<br/>
<b>hence </b><a NAME="E12:49"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="sgraph1.html#R4" title="SGRAPH1:pred.4">is_cycle_of</a>  <a href="sgraph1.html#K8" title="SGRAPH1:func.8">TriangleGraph</a> 
 <b>by </b><i><a class="ref" href="sgraph1.html#D14" target="_self" title="SGRAPH1:def.14">Def14</a></i>;<br/>


</div>
