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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:22"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:22"/><i><font color="Green">E7</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" target="_self">Def1</a></i>;<br/>
<b>assume </b><a NAME="E3:22"/>
<span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan8.html#E5:22"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:22"/>
4 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E8:22"/><b>then </b><i><font color="Green">E10</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E9:22"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan8.html#E2:22"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan8.html#E8:22"><i><font color="Green">E10</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E10:22"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:22"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E12:22"/><i><font color="Green">E12</font></i>: 
2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E13:22"/>
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="jordan8.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E14:22"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E15:22"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan8.html#E12:22"><i><font color="Green">E12</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E16:22"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E17:22"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="txt" href="jordan8.html#E11:22"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E18:22"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E19:22"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan8.html#E1:22"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan8.html#E19:22"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>suppose </b></a><a NAME="E1:22_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="gobrd11.html#T24">GOBRD11:24</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:22_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:22_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E6:22_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan8.html#E4:22_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:22_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan8.html#E17:22"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan8.html#E5:22_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E8:22_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2"><b>suppose </b></a><a NAME="E1:22_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:22_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="gobrd11.html#T27">GOBRD11:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:22_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:22_1_2"/><i><font color="Green">E16</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E6:22_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan8.html#E4:22_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:22_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan8.html#E17:22"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan8.html#E5:22_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E8:22_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_3"><b>suppose </b></a><a NAME="E1:22_1_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:22_1_3"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="gobrd11.html#T30">GOBRD11:30</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:22_1_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:22_1_3"/><i><font color="Green">E16</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a></i>;<br/><a NAME="E6:22_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan8.html#E4:22_1_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:22_1_3"/><b>then </b>
(  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan8.html#E6:22"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan8.html#E17:22"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan8.html#E5:22_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E8:22_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
