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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">p</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> )
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">l</font> = <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4;<br/>
<a NAME="E5:35"/>
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4 <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/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:35"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green">E52</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4
 <b>and </b><br/><a NAME="E9:35"/><i><font color="Green">E54</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4
 <b>by </b><i><a class="ref" href="gobrd14.html#T21">GOBRD14:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<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">E</font> =  <a href="pscomp_1.html#K17">E-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">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"><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>;<br/>
<b>set </b><font color="Maroon">NS</font> = <span class="p1">(<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="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E10:35"/><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:35"/><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>, <font color="Maroon">J</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </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"><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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<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> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a>, </i>;<br/>
<a NAME="E13:35"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</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</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</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:35"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E15:35"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">I</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">J</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">I</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">J</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">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/>
<a NAME="E17:35"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></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">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="E18:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font> <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"><font color="Maroon">I</font> <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="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#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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">J</font> <a href="xcmplx_0.html#K6">-</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">JORDAN8:def 1</a></i>;<br/>
<a NAME="E19:35"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></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</font></span>)</span> <a href="xcmplx_0.html#K2">+</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</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</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E20:35"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<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">p</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:35"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></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">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="E22:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font> <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="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</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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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</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#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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">J</font> <a href="xcmplx_0.html#K6">-</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">JORDAN8:def 1</a></i>;<br/>
<a NAME="E23:35"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></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#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E24:35"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <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">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E25:35"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I</font>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">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="E26:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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"><font color="Maroon">I</font> <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"><span class="p5">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">JORDAN8:def 1</a></i>;<br/>
<a NAME="E27:35"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<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>  <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</font></span>)</span> <a href="xcmplx_0.html#K2">+</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</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</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="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E28:35"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<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="E29:35"/><i><font color="Green">E127</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan1c.html#E2"><i><font color="Green">Lemma42</font></i></a>, , , <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E31:35"/>
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/>
<a NAME="E32:35"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i>, , , , <a class="txt" href="jordan1c.html#E1:35"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E33:35"/><i><font color="Green">E130</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E34:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:35"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E36:35"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">GIJ</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E38:35"/>
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/>
<a NAME="E39:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i>, , , , <a class="txt" href="jordan1c.html#E1:35"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E40:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:35"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E42:35"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">GIJ</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E44:35"/>
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/>
<a NAME="E45:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i>, , , , <a class="txt" href="jordan1c.html#E1:35"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E46:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E47:35"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E49:35"/>
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/>
<a NAME="E50:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i>, , , , <a class="txt" href="jordan1c.html#E1:35"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E51:35"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:35"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E53:35"/><b>then </b><i><font color="Green">E134</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">I</font>
;<br/><b>take </b>
<font color="Maroon">J</font>
;<br/><b>thus </b><a NAME="E2:35_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">I</font> &amp; <font color="Maroon">I</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 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#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan1c.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:35_1_1"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , , , , <a class="txt" href="jordan1c.html#E1:35"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan1c.html#T2" target="_self">Th2</a>, , <a class="txt" href="jordan1c.html#E8:35"><i><font color="Green">E52</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:35_1_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">I</font>,<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="txt" href="jordan1c.html#E2"><i><font color="Green">Lemma42</font></i></a>, , , <a class="ref" href="jordan1c.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font>;<br/><b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E3:35_1_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:35_1_2"/><i><font color="Green">E138</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> &amp; <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:35_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E6:35_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/><a NAME="E7:35_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <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">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"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <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#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="jordan1c.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E8:35_1_2"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan1c.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan9.html#T18">JORDAN9:18</a></i>;<br/><a NAME="E9:35_1_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<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>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<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><a class="txt" href="jordan1c.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan9.html#T18">JORDAN9:18</a></i>;<br/><a NAME="E10:35_1_2"/><b>then </b><i><font color="Green">E144</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E2"><i><font color="Green">Lemma42</font></i></a>, , <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, , , <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E11:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan1c.html#E9"><i><font color="Green">Lemma56</font></i></a>, </i>;<br/><a NAME="E12:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:35_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E14:35_1_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">GIJ</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><b>take </b>
<font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1
;<br/><b>take </b>
<font color="Maroon">J</font>
;<br/><a NAME="E15:35_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E16:35_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
( 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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan1c.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:35_1_2"/>
<font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_2_1"/>
<font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:35_1_2_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E3:35_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan8.html#T14">JORDAN8:14</a></i>;<br/>
<a NAME="E4:35_1_2_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<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#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1c.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E5:35_1_2_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:35_1_2_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<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#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:35_1_2_1"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T114">JORDAN2C:114</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_2_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:35_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 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#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan1c.html#E1"><i><font color="Green">Lemma40</font></i></a>, , <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan1c.html#E10"><i><font color="Green">Lemma57</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E20:35_1_2"/><i><font color="Green">E149</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">J</font> &amp; <font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E21:35_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">GIJ</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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E22:35_1_2"/>
<font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/><a NAME="E23:35_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <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">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"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <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#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="ref" href="jordan1c.html#T3" target="_self">Th3</a>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><a NAME="E24:35_1_2"/><b>then </b>
<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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E2"><i><font color="Green">Lemma42</font></i></a>, , , , , <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E25:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E26:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E4"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:35_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">GIJ</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E28:35_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">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"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="jordan1c.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan1c.html#E8:35"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E29:35_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">P</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , , , , , </i>;<br/><b>hence </b><a NAME="E30:35_1_2"/>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">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"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="txt" href="jordan1c.html#E2"><i><font color="Green">Lemma42</font></i></a>, , , , , <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
