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

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</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">G</font> =  <a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">j</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</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E50</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</font>,<font color="Maroon">n</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">n</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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:21"/>
<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</font>,<font color="Maroon">n</font></span>)</span>,<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</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</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">p</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="E5:21"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</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</font>,<font color="Maroon">n</font></span>)</span>,<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</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:21"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">p</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:21"/>
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</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green">E56</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</font>,<font color="Maroon">n</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">W</font> =  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">EW</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span>;<br/>
<a NAME="E9:21"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><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</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,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</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,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</font></span>)</span> <a href="xcmplx_0.html#K2">+</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</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</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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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</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</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</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">n</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/>;<br/>
<a NAME="E11:21"/><i><font color="Green">E58</font></i>: 
2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</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="E12:21"/>
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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="E14:21"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E15:21"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><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</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></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"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E17:21"/><b>then </b>
<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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</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"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</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">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E18:21"/><b>then </b><i><font color="Green">E60</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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E19:21"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</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="E20:21"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</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="21_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">j</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</font>,<font color="Maroon">n</font></span>)</span> or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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</font>,<font color="Maroon">n</font></span>)</span> ) )
 </span><b>by </b><i>, , <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="21_2_1"><b>suppose </b></a><a NAME="E1:21_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_2_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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <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">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> 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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <font color="Olive">s</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</font>,<font color="Maroon">n</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">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:21_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green">E63</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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <font color="Maroon">s</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</font>,<font color="Maroon">n</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/>;<br/><a NAME="E6:21_2_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:21_2_1"/><b>then </b>
(  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , <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="E8:21_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2"><b>suppose </b></a><a NAME="E1:21_2_2"/>
<font color="Maroon">j</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</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_2_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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <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">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> 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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="gobrd11.html#T28">GOBRD11:28</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:21_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:21_2_2"/><i><font color="Green">E68</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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><a NAME="E6:21_2_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:21_2_2"/><b>then </b>
(  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , <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="E8:21_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3"><b>suppose </b></a><a NAME="E1:21_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:21_2_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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" target="_self">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <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">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> 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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="gobrd11.html#T29">GOBRD11:29</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:21_2_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:21_2_3"/><i><font color="Green">E70</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</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</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">r</font> &amp; <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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</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</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i/>;<br/><a NAME="E6:21_2_3"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:21_2_3"/><b>then </b>
(  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , <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="E8:21_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
