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

<b>let </b><font color="Maroon">i</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">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> ) or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,0,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> <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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></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#T25">GOBRD11:25</a></i>;<br/><a NAME="E3:29_1_1"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">q</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> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,0,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>take </b><font color="Maroon">q</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>

<a NAME="E1:29_1_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<b>hence </b><a NAME="E2:29_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,0,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/>

<a NAME="E3:29_1_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_1_1"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:29_1_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E4:29_1_1_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:29_1_1_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E6:29_1_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T8">TOPREAL6:8</a></i>;<br/><a NAME="E7:29_1_1_1_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E8:29_1_1_1_1_2"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:29_1_1"/>
not  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T40">JORDAN2C:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:29_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> <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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,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">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <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>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></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#T31">GOBRD11:31</a></i>;<br/><a NAME="E3:29_1_2"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">q</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> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>take </b><font color="Maroon">q</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>

<a NAME="E1:29_1_2_1"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E2:29_1_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:29_1_2_1"/>
( <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">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:29_1_2_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <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>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/>
<a NAME="E5:29_1_2_1"/>
 <a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<a NAME="E7:29_1_2_1"/><i><font color="Green">E98</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_2_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:29_1_2_1_1_1"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_2_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:29_1_2_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E4:29_1_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T7">TOPREAL6:7</a></i>;<br/><a NAME="E5:29_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_1_2_1_1_2"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:29_1_2"/>
not  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40">JORDAN2C:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3"><b>suppose </b></a><a NAME="E1:29_1_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:29_1_3"/><i><font color="Green">E102</font></i>: 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></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/><a NAME="E3:29_1_3"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  ex <font color="Olive">q</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> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>take </b><font color="Maroon">q</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>

<a NAME="E1:29_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E2:29_1_3_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:29_1_3_1"/><i><font color="Green">E103</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_3_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_3_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_3_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:29_1_3_1_1_1"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T50">JGRAPH_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_3_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_3_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:29_1_3_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E4:29_1_3_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E5:29_1_3_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_1_3_1_1_2"/>
not <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:29_1_3"/>
not  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40">JORDAN2C:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
