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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) ) &amp; <font color="Olive">f</font> is <a href="topreal1.html#V1">special</a> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">f</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/><font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) holds <br/> ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> );<br/>
<a NAME="E1:19"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</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="E1:19_1"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) ) &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 ;<br/>

<b>take </b><font color="Maroon">g</font> = <font color="Maroon">f</font>;<br/>

<a NAME="E2:19_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
( <font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:19_2"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:19_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>and </b><br/><a NAME="E4:19_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a>
 <b>and </b><br/><a NAME="E5:19_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E6:19_2"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">1</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">f</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/><a NAME="E2:19_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_1_1"/>
for <font color="Olive">i1</font>, <font color="Olive">i2</font>, <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j1</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">i2</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">j1</font>,<font color="Olive">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="real_1.html#K5">-</a> <font color="Olive">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:19_2_1_1_1"/>
( <font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="E3:19_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</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">f</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E2:19_2_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:19_2_1_1_2_1"/><i><font color="Green">E73</font></i>: 
( <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> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:19_2_1_1_2_1"/>
( <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> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><a NAME="E6:19_2_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> is <a href="topreal1.html#V1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a><br/>

<b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:19_2_1_1_2_2"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 ;<br/>

<a NAME="E2:19_2_1_1_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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="E3:19_2_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:19_2_1_1_2_2"/><b>then </b><i><font color="Green">E101</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E6:19_2_1_1_2_2"/>
<font color="Maroon">n</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">f</font>
 
<b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:19_2_1_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <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"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</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"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 ;<br/><a NAME="E2:19_2_1_1_2_3"/><b>then </b><i><font color="Green">E103</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><a NAME="E3:19_2_1_1_2_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E9:19_2_1_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">g1</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 <b>by </b><i>, , , </i>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:19_2_1_1_2"/><i><font color="Green">E105</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:19_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:19_2_1_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:19_2_1_1_2"/><i><font color="Green">E107</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i/>;<br/><a NAME="E16:19_2_1_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><a NAME="E17:19_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E18:19_2_1_1_2"/><i><font color="Green">E110</font></i>: 
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> holds <br/> ex <font color="Olive">m</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> ) ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E19:19_2_1_1_2"/><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">l1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>, <font color="Maroon">c1</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">i2</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon">x1</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font>;<br/><b>set </b><font color="Maroon">y1</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font>;<br/><b>set </b><font color="Maroon">x2</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font>;<br/><b>set </b><font color="Maroon">y2</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font>;<br/><a NAME="E21:19_2_1_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D6">GOBOARD1:def 6</a></i>;<br/><a NAME="E22:19_2_1_1_2"/><i><font color="Green">E118</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font> is <a href="goboard1.html#V1">increasing</a>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D8">GOBOARD1:def 8</a></i>;<br/><a NAME="E23:19_2_1_1_2"/><i><font color="Green">E119</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font> is <a href="goboard1.html#V1">increasing</a>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D9">GOBOARD1:def 9</a></i>;<br/><a NAME="E24:19_2_1_1_2"/><i><font color="Green">E120</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D7">GOBOARD1:def 7</a></i>;<br/><a NAME="E25:19_2_1_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:19_2_1_1_2"/><i><font color="Green">E122</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E27:19_2_1_1_2"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E28:19_2_1_1_2"/><i><font color="Green">E124</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E29:19_2_1_1_2"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E30:19_2_1_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/><a NAME="E31:19_2_1_1_2"/><i><font color="Green">E127</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E32:19_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E33:19_2_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E34:19_2_1_1_2"/><i><font color="Green">E129</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> )
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 </span><b>by </b><i>, , , , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">ppi</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>;<br/><b>set </b><font color="Maroon">pj</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_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="E1:19_2_1_1_2_4_1_1_1_1"/>
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> or <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E10:19_2_1_1_2_4_1_1_1_1_1_1"/>
( <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></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> <font color="Maroon">G</font> &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Olive">m</font>;<br/><div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a></i>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">p</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1"/><i><font color="Green">E139</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>();<br/></i><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1"/><i><font color="Green">E140</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_3"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">i1</font>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E142</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_4"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E145</font></i>: 
( <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></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> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>] &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>, <font color="Maroon">m2</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_4"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">m1</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">m2</font> )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a></i>;<br/><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> )
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_4"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E151</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_5"/><i><font color="Green">E152</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_5"/><b>then </b><i><font color="Green">E153</font></i>: 
( <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</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> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[1,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1] &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_5"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">m1</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_5"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_1_1_1_1"/>
<font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon">lk</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">w</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_1"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , , , , , , , , <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green">E158</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">w</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
<b>by </b><i>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>
;<br/><b>thus </b><a NAME="E20:19_2_1_1_2_4_1_1_1_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7"/><i><font color="Green">E160</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E161</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1"/><i><font color="Green">E162</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <font color="Maroon">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<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="E3:19_2_1_1_2_4_1_1_1_1_1_7_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_1"/><i><font color="Green">E164</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E166</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, </i>;<br/><b>thus </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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>  &amp; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , , , , <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E168</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E172</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
( <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">w</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font>] )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b>
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">u</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">u</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , , , , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E15:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_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="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_2"/>
( <font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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> 
 ;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E19:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E20:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E21:19_2_1_1_2_4_1_1_1_1_1_7_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</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>  &amp; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_2"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &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="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1"/>
( <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">g1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><i><font color="Green">E180</font></i>: 
<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">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
( <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">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
( 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> <font color="Maroon">g1</font> &amp; 1 <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">g1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal3.html#T25">TOPREAL3:25</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="topreal3.html#T27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><i><font color="Green">E183</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;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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="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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">q2</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><b>then </b><i><font color="Green">E187</font></i>: 
( <font color="Maroon">q1</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">q2</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/><i><font color="Green">E188</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_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="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</font> is   <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> : ( <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; ( for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  ) );<br/><div><i><font color="Green">E194</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>;<br/><b>thus </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <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="real_2.html#T173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon">q</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="E2:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_1_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">ma</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">ma</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">ma</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="E2:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E200</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E201</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><i><font color="Green">E202</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font>
<b>by </b><i>, , <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>
;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</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"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <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="E10:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E205</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E15:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E210</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">m1</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E19:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> as   <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/><a NAME="E21:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E22:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">q</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"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E216</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon">lq</font> = <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><b>then </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E26:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E27:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, , <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><a NAME="E28:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E29:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E30:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">ma</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> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E219</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>, <font color="Maroon">qa1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><div><i><font color="Green">E223</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2_1"/><b>then </b>
for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
( <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">lma</font> = <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">qa</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"><font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">qa1</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"><font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> = 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">qa1</font>,<font color="Maroon">qa</font>
<b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_1_7_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2_1_1_2_4_1_1_1_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E21:19_2_1_1_2_4_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E24:19_2_1_1_2_4_1_1_1_1_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E227</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E26:19_2_1_1_2_4_1_1_1_1_1"/><b>then </b><i><font color="Green">E228</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">m1</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>thus </b><a NAME="E29:19_2_1_1_2_4_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_2"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_2"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3"/><i><font color="Green">E230</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> holds <br/><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" target="_self">GOBOARD2:sch 1</a>();<br/></i><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3"/><i><font color="Green">E231</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g2</font>;<br/><div><i><font color="Green">E232</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_1"/><b>then </b><i><font color="Green">E233</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <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="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></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> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3"/><i><font color="Green">E234</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">i1</font>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E236</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E237</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_3"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_3"/><b>then </b><i><font color="Green">E238</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></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> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> )
 ;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_3"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E239</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_4"/><i><font color="Green">E240</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_4"/><b>then </b>
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</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> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_4"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_1_1_1_3"/>
<font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon">lk</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">w</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_3"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b>
( <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E241</font></i>: 
( <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E242</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E243</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , , , , , , , , <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green">E244</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">w</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
<b>by </b><i>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>
;<br/><b>thus </b><a NAME="E20:19_2_1_1_2_4_1_1_1_1_3"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6"/><i><font color="Green">E245</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E246</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1"/><i><font color="Green">E247</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <font color="Maroon">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<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="E3:19_2_1_1_2_4_1_1_1_1_3_6_1"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><b>set </b><font color="Maroon">u</font> = <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_1"/><i><font color="Green">E249</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E251</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, </i>;<br/><b>thus </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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>  &amp; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , , , , <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E253</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E258</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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="E10:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E260</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <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><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , </i>;<br/><a NAME="E13:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">l1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E263</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">l1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , , , , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E15:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><b>hence </b><a NAME="E19:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_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="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1"/>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> or <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_1"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E21:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><b>thus </b><a NAME="E22:19_2_1_1_2_4_1_1_1_1_3_6_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</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>  &amp; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">l1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_2"/><i><font color="Green">E265</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &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="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1"/>
( <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">g1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><i><font color="Green">E267</font></i>: 
<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">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
( <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">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
( 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> <font color="Maroon">g1</font> &amp; 1 <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">g1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal3.html#T25">TOPREAL3:25</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="topreal3.html#T27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><b>then </b><i><font color="Green">E269</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><i><font color="Green">E270</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;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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="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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">q2</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
( <font color="Maroon">q1</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">q2</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/><i><font color="Green">E273</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_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="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/>
<font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</font> is   <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> : ( <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/><i><font color="Green">E275</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6"/><b>then </b><i><font color="Green">E276</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"/><i><font color="Green">E277</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; ( for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  ) );<br/><div><i><font color="Green">E278</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>;<br/><b>thus </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <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="real_2.html#T173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon">q</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="E2:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_1_1"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"/><i><font color="Green">E280</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">ma</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"/><i><font color="Green">E281</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">ma</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">ma</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="E2:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E283</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E284</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E285</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>
;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E286</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</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"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E287</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <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="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <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="E10:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E288</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E289</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E15:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E292</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E293</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">m1</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E19:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E295</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> as   <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/><a NAME="E21:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E296</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E22:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">q</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"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E299</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon">lq</font> = <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><b>then </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E26:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E27:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K22">`2</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> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, , <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>;<br/><a NAME="E28:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E29:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E30:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">ma</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> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E300</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>, <font color="Maroon">qa1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><div><i><font color="Green">E302</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2_1"/>
<font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2_1"/><b>then </b>
for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E3:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E303</font></i>: 
( <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">lma</font> = <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> ;<br/><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><i><font color="Green">E304</font></i>: 
( <font color="Maroon">qa</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"><font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">qa1</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"><font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> = 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">qa</font>,<font color="Maroon">qa1</font>
<b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="topreal3.html#T15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_1_1_1_3_6_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2_1_1_2_4_1_1_1_1_3_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E21:19_2_1_1_2_4_1_1_1_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E24:19_2_1_1_2_4_1_1_1_1_3"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E305</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E26:19_2_1_1_2_4_1_1_1_1_3"/><b>then </b><i><font color="Green">E306</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/><b>thus </b><a NAME="E28:19_2_1_1_2_4_1_1_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_1"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">ppi</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>;<br/><b>set </b><font color="Maroon">pj</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_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="E1:19_2_1_1_2_4_1_2_1_1"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j1</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><div><i><font color="Green">E309</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E310</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:19_2_1_1_2_4_1_2_1_1_1_1"/>
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Maroon">i2</font>;<br/><div><i><font color="Green">E311</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a></i>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i2</font>;<br/><b>thus </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">p</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1"/><i><font color="Green">E312</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>();<br/></i><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1"/><i><font color="Green">E313</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_3"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">i2</font>;<br/><b>thus </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E315</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E316</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_4"/><i><font color="Green">E317</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_4"/><b>then </b><i><font color="Green">E318</font></i>: 
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>] &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>, <font color="Maroon">m2</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_4"/><i><font color="Green">E319</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m2</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a></i>;<br/><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> )
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_4"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E320</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_5"/><i><font color="Green">E321</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_5"/><b>then </b><i><font color="Green">E322</font></i>: 
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[1,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1] &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_5"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m1</font>,<font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_5"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_2_1_1_1"/>
<font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon">lk</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">w</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><a NAME="E14:19_2_1_1_2_4_1_2_1_1_1"/>
( <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E323</font></i>: 
( <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E324</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E325</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , , , , , <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green">E326</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">w</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
<b>by </b><i>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>
;<br/><b>thus </b><a NAME="E19:19_2_1_1_2_4_1_2_1_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7"/><i><font color="Green">E327</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E328</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1"/><i><font color="Green">E329</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <font color="Maroon">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<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="E3:19_2_1_1_2_4_1_2_1_1_1_7_1"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E332</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>thus </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><i><font color="Green">E333</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E334</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E335</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><i><font color="Green">E336</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E337</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E338</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E340</font></i>: 
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">w</font>,<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font>] )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">w</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E341</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">u</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">u</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b><i><font color="Green">E342</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">u</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">u</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E14:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , , <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_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="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_2"/><i><font color="Green">E343</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_2"/>
( <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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> 
 ;<br/><a NAME="E17:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E19:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E20:19_2_1_1_2_4_1_2_1_1_1_7_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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="relat_1.html#K2">rng</a> <font color="Maroon">c1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_2"/><i><font color="Green">E344</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2"/><i><font color="Green">E345</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &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="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1"/>
( <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">g1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><i><font color="Green">E346</font></i>: 
<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">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
( <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">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
( 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> <font color="Maroon">g1</font> &amp; 1 <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">g1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal3.html#T25">TOPREAL3:25</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="topreal3.html#T27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><b>then </b><i><font color="Green">E348</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><i><font color="Green">E349</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;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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="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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">q2</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><i><font color="Green">E350</font></i>: 
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><b>then </b><i><font color="Green">E351</font></i>: 
( <font color="Maroon">q1</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">q2</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"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/><i><font color="Green">E352</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_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="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/><i><font color="Green">E353</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</font> is   <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> : ( <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/><i><font color="Green">E354</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7"/><b>then </b><i><font color="Green">E355</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"/><i><font color="Green">E356</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; ( for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  ) );<br/><div><i><font color="Green">E357</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>;<br/><b>thus </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <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="real_2.html#T173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon">q</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="E2:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"/><i><font color="Green">E358</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_1_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"/><i><font color="Green">E359</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">ma</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"/><i><font color="Green">E360</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">ma</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">ma</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="E2:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/>
<font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E362</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E363</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><i><font color="Green">E364</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>
;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">p1</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E365</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</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"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E366</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <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="E10:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E367</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E368</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E369</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E370</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E15:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E371</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E372</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E373</font></i>: 
<font color="Maroon">m1</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E19:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E374</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> as   <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/><a NAME="E21:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E22:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E376</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <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="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">q</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"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><i><font color="Green">E378</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon">lq</font> = <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><b>then </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E26:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E27:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, , <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><a NAME="E28:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E29:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E30:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">ma</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> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E379</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>, <font color="Maroon">qa1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><div><i><font color="Green">E381</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2_1"/><b>then </b>
for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E382</font></i>: 
( <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">lma</font> = <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><i><font color="Green">E383</font></i>: 
( <font color="Maroon">qa</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"><font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">qa1</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"><font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> = 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">qa1</font>,<font color="Maroon">qa</font>
<b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_1_7_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2_1_1_2_4_1_2_1_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E20:19_2_1_1_2_4_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E23:19_2_1_1_2_4_1_2_1_1_1"/>
<font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E384</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E25:19_2_1_1_2_4_1_2_1_1_1"/><b>then </b><i><font color="Green">E385</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>thus </b><a NAME="E28:19_2_1_1_2_4_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_2"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3"><b>case </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3"/><i><font color="Green">E386</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">i2</font>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3"/><i><font color="Green">E387</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> holds <br/><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" target="_self">GOBOARD2:sch 1</a>();<br/></i><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3"/><i><font color="Green">E388</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g2</font>;<br/><div><i><font color="Green">E389</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_1"/><b>then </b><i><font color="Green">E390</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_1"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3"/><i><font color="Green">E391</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_2"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">i2</font>;<br/><b>thus </b><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E393</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E394</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_3"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_3"/><b>then </b><i><font color="Green">E395</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">i2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> )
 ;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_3"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E396</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l4</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_4"/><i><font color="Green">E397</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l3</font>,<font color="Maroon">l4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">l2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l3</font>,<font color="Maroon">l4</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_4"/><b>then </b>
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_4"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">l3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">l4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_2_1_1_3"/>
<font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon">lk</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">w</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><a NAME="E14:19_2_1_1_2_4_1_2_1_1_3"/>
( <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E398</font></i>: 
( <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E399</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E400</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , , , , , <a class="ref" href="euclid.html#T57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green">E401</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <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> : ( <font color="Olive">w</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">w</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
<b>by </b><i>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>
;<br/><b>thus </b><a NAME="E19:19_2_1_1_2_4_1_2_1_1_3"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6"/><i><font color="Green">E402</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E403</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1"/><i><font color="Green">E404</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <font color="Maroon">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<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="E3:19_2_1_1_2_4_1_2_1_1_3_6_1"/><i><font color="Green">E405</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><b>set </b><font color="Maroon">u</font> = <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/><i><font color="Green">E406</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E407</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>thus </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , , , <a class="txt" href="goboard2.html#E2:19_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><i><font color="Green">E408</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E409</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><i><font color="Green">E410</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E411</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E413</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E414</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E415</font></i>: 
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E10:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b>
<font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b><i><font color="Green">E417</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</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">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</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">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font></span>)</span>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , <a class="txt" href="goboard2.html#E1:19_2"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E15:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E16:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><b>hence </b><a NAME="E17:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_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="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_1"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_1"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_2"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_2"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
<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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><b>thus </b><a NAME="E20:19_2_1_1_2_4_1_2_1_1_3_6_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c1</font>
 <b>by </b><i>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</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; <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"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</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="relat_1.html#K2">rng</a> <font color="Maroon">c1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_2"/><i><font color="Green">E419</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2"/><i><font color="Green">E420</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &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="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1"/>
( <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">g1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><i><font color="Green">E421</font></i>: 
<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">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
( <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">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
( 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> <font color="Maroon">g1</font> &amp; 1 <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">g1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal3.html#T25">TOPREAL3:25</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g1</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="topreal3.html#T27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><i><font color="Green">E422</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><b>then </b><i><font color="Green">E423</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <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> <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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><i><font color="Green">E424</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;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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="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="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">q2</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><i><font color="Green">E425</font></i>: 
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q2</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><b>then </b><i><font color="Green">E426</font></i>: 
( <font color="Maroon">q1</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"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">q2</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"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/><i><font color="Green">E427</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_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="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/><i><font color="Green">E428</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/>
<font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</font> is   <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> : ( <font color="Olive">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/><i><font color="Green">E429</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</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">f</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6"/><b>then </b><i><font color="Green">E430</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard2.html#E4:19_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"/><i><font color="Green">E431</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:19_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; ( for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  ) );<br/><div><i><font color="Green">E432</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>;<br/><b>thus </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <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="real_2.html#T173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon">q</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="E2:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_1_1"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"/><i><font color="Green">E434</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">ma</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"/><i><font color="Green">E435</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">ma</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="goboard2.html#E6:19_2"><i><font color="Green">E65</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E436</font></i>: 
<font color="Maroon">ma</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="E2:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E437</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E438</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E439</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>
;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E440</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</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"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E441</font></i>: 
( <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <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="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <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="E10:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E442</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E443</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard2.html#E3"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E13:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E444</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E445</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><a NAME="E15:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E446</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E447</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E448</font></i>: 
<font color="Maroon">m1</font> <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="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E19:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E449</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> as   <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/><a NAME="E21:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E450</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E22:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E451</font></i>: 
<font color="Maroon">q</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="goboard2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E23:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E452</font></i>: 
<font color="Maroon">q</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"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><i><font color="Green">E453</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon">lq</font> = <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">e</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><b>then </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="goboard2.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="goboard2.html#E2:19"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard2.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E26:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E27:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is   <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> : ( <font color="Olive">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">e</font> <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> <font color="Maroon">j1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T11" target="_self">Th11</a>, , <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>;<br/><a NAME="E28:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="goboard2.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard2.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E29:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E30:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/>
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">ma</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> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E454</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon">qa</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>, <font color="Maroon">qa1</font> = <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</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#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><i><font color="Green">E455</font></i>: 
<font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><div><i><font color="Green">E456</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2_1"/>
<font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2_1"/><b>then </b>
for <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 <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E3:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E457</font></i>: 
( <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="goboard2.html#E1:19_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">lma</font> = <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> ;<br/><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><i><font color="Green">E458</font></i>: 
( <font color="Maroon">qa</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"><font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">qa1</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"><font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">qa1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> = 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">qa</font>,<font color="Maroon">qa1</font>
<b>by </b><i><a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qa1</font> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard2.html#E5:19_2_1_1_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="topreal3.html#T16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">ma</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard2.html#E5:19_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard2.html#E2:19_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard2.html#E3:19_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard2.html#E4:19_2_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E12:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E13:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:19_2_1_1_2_4_1_2_1_1_3_6_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2_1_1_2_4_1_2_1_1_3_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E20:19_2_1_1_2_4_1_2_1_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E23:19_2_1_1_2_4_1_2_1_1_3"/>
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E24:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E459</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E25:19_2_1_1_2_4_1_2_1_1_3"/><b>then </b><i><font color="Green">E460</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>,<font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="goboard2.html#E1:19"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/><b>thus </b><a NAME="E27:19_2_1_1_2_4_1_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_2_1_1_2_4_1_2"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E36:19_2_1_1_2"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_2"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:19"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) ) &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) implies  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">g</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> ) )
 ;<br/>


</div>
