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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> ) ) &amp; <font color="Olive" title="b1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b2">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b2">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b2">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b2">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> );<br/>
<a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green" title="E4">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> ) ) &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> )
 ;<br/>

<b>take </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font>;<br/>

<a NAME="E2:2_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1"/>
( <font color="Maroon" title="c4">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green" title="E4">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:2_2"/><i><font color="Green" title="E6">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E3:2_2"/><i><font color="Green" title="E7">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>and </b><br/><a NAME="E4:2_2"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:2_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>and </b><br/><a NAME="E6:2_2"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>and </b><br/><a NAME="E7:2_2"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_1"/><b>then </b><i><font color="Green" title="E12">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> )
 ;<br/><a NAME="E2:2_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_1_1"/>
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:2_2_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>thus </b><a NAME="E5:2_2_1_1"/>
( <font color="Maroon" title="c5">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2"/><i><font color="Green" title="E12">A12</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2"/><b>then </b><i><font color="Green" title="E13">A13</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:2_2_1_2"/><b>then </b><i><font color="Green" title="E14">A14</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:2_2_1_2"/><b>then </b><i><font color="Green" title="E15">A15</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2"/><i><font color="Green" title="E16">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E6:2_2_1_2"/><i><font color="Green" title="E17">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:2_2_1_2"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:2_2_1_2"/><i><font color="Green" title="E19">A19</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font>;<br/><div><i><font color="Green" title="E20">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_1"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><a NAME="E2:2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:2_2_1_2_1"/><i><font color="Green" title="E22">A22</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c7">i</font>;<br/><b>take </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c8">j</font>;<br/><b>thus </b><a NAME="E5:2_2_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><a NAME="E10:2_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E11:2_2_1_2"/><b>then </b><i><font color="Green" title="E21">A23</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/><a NAME="E12:2_2_1_2"/><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="goboard3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E13:2_2_1_2"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T12" title="GOBOARD2:th.12">GOBOARD2:12</a></i>;<br/><a NAME="E14:2_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_2"/><i><font color="Green" title="E24">A26</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 ;<br/>

<a NAME="E2:2_2_1_2_2"/><b>then </b>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E3:2_2_1_2_2"/><b>then </b><i><font color="Green" title="E25">A27</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E4:2_2_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E5:2_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_2"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_2"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_2"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c6">g1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2"/><i><font color="Green" title="E24">A28</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E17:2_2_1_2"/><i><font color="Green" title="E25">A29</font></i>: 
<font color="Maroon" title="c6">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E18:2_2_1_2"/><i><font color="Green" title="E26">A30</font></i>: 
<font color="Maroon" title="c6">g1</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>and </b><br/><a NAME="E19:2_2_1_2"/><i><font color="Green" title="E27">A31</font></i>: 
<font color="Maroon" title="c6">g1</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>and </b><br/><a NAME="E20:2_2_1_2"/><i><font color="Green" title="E28">A32</font></i>: 
<font color="Maroon" title="c6">g1</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>and </b><br/><a NAME="E21:2_2_1_2"/><i><font color="Green" title="E29">A33</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>and </b><br/><a NAME="E22:2_2_1_2"/><i><font color="Green" title="E30">A34</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>and </b><br/><a NAME="E23:2_2_1_2"/><i><font color="Green" title="E31">A35</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E24:2_2_1_2"/><i><font color="Green" title="E32">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2"><i><font color="Green" title="E23">A25</font></i></a></i>;<br/><a NAME="E25:2_2_1_2"/><i><font color="Green" title="E33">A37</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies (  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_3"/><i><font color="Green" title="E34">A38</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>hence </b><a NAME="E2:2_2_1_2_3"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="topreal1.html#T28" title="TOPREAL1:th.28">TOPREAL1:28</a></i>;<br/>

<a NAME="E3:2_2_1_2_3"/><b>then </b><i><font color="Green" title="E35">A39</font></i>: 
( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_3"><i><font color="Green" title="E34">A38</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="topreal1.html#T28" title="TOPREAL1:th.28">TOPREAL1:28</a></i>;<br/>
<a NAME="E4:2_2_1_2_3"/><b>then </b><i><font color="Green" title="E36">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:2_2_1_2_3"/><i><font color="Green" title="E37">A41</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E6:2_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_3"><i><font color="Green" title="E36">A40</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E7:2_2_1_2_3"/><b>then </b><i><font color="Green" title="E38">A42</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:2_2_1_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_3_1"/><i><font color="Green" title="E39">A43</font></i>: 
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_3_1"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_3_1"><i><font color="Green" title="E39">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:2_2_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_3"><i><font color="Green" title="E35">A39</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_3"><i><font color="Green" title="E37">A41</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_3_1"><i><font color="Green" title="E39">A43</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:2_2_1_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_3"><i><font color="Green" title="E38">A42</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:2_2_1_2"/><i><font color="Green" title="E34">A44</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_4"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/>

<a NAME="E2:2_2_1_2_4"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:2_2_1_2_4"/><b>then </b><i><font color="Green" title="E35">A45</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_2_1_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:2_2_1_2_4"/><i><font color="Green" title="E36">A46</font></i>: 
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>thus </b><a NAME="E7:2_2_1_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_4"><i><font color="Green" title="E35">A45</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_4"><i><font color="Green" title="E36">A46</font></i></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c8">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E28:2_2_1_2"/><i><font color="Green" title="E35">A47</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><a NAME="E29:2_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E30:2_2_1_2"/><b>then </b><i><font color="Green" title="E36">A48</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:2_2_1_2"/><i><font color="Green" title="E37">A49</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c10">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/><a NAME="E33:2_2_1_2"/><i><font color="Green" title="E38">A50</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/><a NAME="E34:2_2_1_2"/><b>then </b><i><font color="Green" title="E39">A51</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E35:2_2_1_2"/><i><font color="Green" title="E40">A52</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E36:2_2_1_2"/><i><font color="Green" title="E41">A53</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> holds <br/> ex <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="goboard3.html#E16:2_2_1_2"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">l1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c12">c1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c8">i2</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c13">x1</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font>;<br/><b>set </b><font color="Maroon" title="c14">y1</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font>;<br/><b>set </b><font color="Maroon" title="c15">x2</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font>;<br/><b>set </b><font color="Maroon" title="c16">y2</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font>;<br/><a NAME="E38:2_2_1_2"/><i><font color="Green" title="E42">A54</font></i>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="goboard1.html#D6" title="GOBOARD1:def.6">GOBOARD1:def 6</a></i>;<br/><a NAME="E39:2_2_1_2"/><i><font color="Green" title="E43">A55</font></i>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="goboard1.html#D8" title="GOBOARD1:def.8">GOBOARD1:def 8</a></i>;<br/><a NAME="E40:2_2_1_2"/><i><font color="Green" title="E44">A56</font></i>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></i>;<br/><a NAME="E41:2_2_1_2"/><i><font color="Green" title="E45">A57</font></i>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="goboard1.html#D7" title="GOBOARD1:def.7">GOBOARD1:def 7</a></i>;<br/><a NAME="E42:2_2_1_2"/><i><font color="Green" title="E46">A58</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E43:2_2_1_2"/><i><font color="Green" title="E47">A59</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E44:2_2_1_2"/><i><font color="Green" title="E48">A60</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E45:2_2_1_2"/><i><font color="Green" title="E49">A61</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/><a NAME="E46:2_2_1_2"/><i><font color="Green" title="E50">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E47:2_2_1_2"/><b>then </b><i><font color="Green" title="E51">A63</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E48:2_2_1_2"/><i><font color="Green" title="E52">A64</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:2_2_1_2"/><i><font color="Green" title="E53">A65</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E50:2_2_1_2"/><i><font color="Green" title="E54">A66</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E55">A67</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">c1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E52:2_2_1_2"/><i><font color="Green" title="E56">A68</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E53:2_2_1_2"/><i><font color="Green" title="E57">A69</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green" title="E58">A70</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">c1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1"/>
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> or <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E7:2_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E30:2_2_1_2"><i><font color="Green" title="E36">A48</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="ref" href="goboard2.html#T16" title="GOBOARD2:th.16">GOBOARD2:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1"/><i><font color="Green" title="E59">A71</font></i>: 
<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c17">ppi</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>;<br/><b>set </b><font color="Maroon" title="c18">pj</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_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="E1:2_2_1_2_7_1_1_1_1"/>
( <font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j2</font> or <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> or <font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j2</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E60">A72</font></i>: 
<font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">j2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><i><font color="Green" title="E61">A73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E62">A74</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_1"><i><font color="Green" title="E62">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">w</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_1"><i><font color="Green" title="E62">A74</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w</font> &amp; <font color="Maroon" title="c21">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_1_1"/>
( <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Olive" title="b1">m</font>;<br/><div><i><font color="Green" title="E62">A75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_2"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c22">p</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c21">m</font>;<br/><b>thus </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">n</font>,<font color="Maroon" title="c22">p</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E63">A76</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_4.html#S1" title="FINSEQ_4:sch.1">FINSEQ_4:sch 1</a>(<a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E62">A75</font></i></a>);<br/></i><a NAME="E7:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E64">A77</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_3"/><i><font color="Green" title="E65">A78</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c7">i1</font>;<br/><b>take </b><font color="Maroon" title="c25">m</font> = <font color="Maroon" title="c23">m</font>;<br/><b>thus </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_3"><i><font color="Green" title="E65">A78</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E65">A79</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E66">A80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_4"/><i><font color="Green" title="E67">A81</font></i>: 
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E68">A82</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font>, <font color="Maroon" title="c24">m2</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_4"><i><font color="Green" title="E67">A81</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_4"/><i><font color="Green" title="E69">A83</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c23">m1</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c24">m2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_4"><i><font color="Green" title="E67">A81</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c25">l1</font>, <font color="Maroon" title="c26">l2</font>, <font color="Maroon" title="c27">l3</font>, <font color="Maroon" title="c28">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l1</font>,<font color="Maroon" title="c26">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">l1</font>,<font color="Maroon" title="c26">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">l4</font> )
 ;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c25">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c26">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">m1</font> &amp; <font color="Maroon" title="c27">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c28">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">m2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_4"><i><font color="Green" title="E68">A82</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_4"><i><font color="Green" title="E69">A83</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">l4</font></span>)</span></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">l1</font>, <font color="Maroon" title="c23">l2</font>, <font color="Maroon" title="c24">l3</font>, <font color="Maroon" title="c25">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_5"/><i><font color="Green" title="E67">A84</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_5"/><b>then </b><i><font color="Green" title="E68">A85</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_5"><i><font color="Green" title="E67">A84</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_5"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c26">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_5"><i><font color="Green" title="E67">A84</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_5"/><b>then </b>
( <font color="Maroon" title="c22">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c23">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c24">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c25">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_5"><i><font color="Green" title="E67">A84</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_5"><i><font color="Green" title="E68">A85</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l4</font></span>)</span></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:2_2_1_2_7_1_1_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E66">A80</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E65">A79</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon" title="c22">lk</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E14:2_2_1_2_7_1_1_1_1_1"/>
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E67">A86</font></i>: 
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E68">A87</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E69">A88</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E38:2_2_1_2"><i><font color="Green" title="E42">A54</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green" title="E70">A89</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E69">A88</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E19:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E71">A90</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_7"/><i><font color="Green" title="E72">A91</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_7"/><b>then </b><i><font color="Green" title="E73">A92</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_7"><i><font color="Green" title="E72">A91</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_7"/><i><font color="Green" title="E74">A93</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c26">m1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_7"><i><font color="Green" title="E72">A91</font></i></a></i>;<br/><b>assume </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_7"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_7"><i><font color="Green" title="E73">A92</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_7"><i><font color="Green" title="E74">A93</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_7"><i><font color="Green" title="E72">A91</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E21:2_2_1_2_7_1_1_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E22:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E72">A94</font></i>: 
( <font color="Maroon" title="c20">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/><a NAME="E23:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E73">A95</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_8"/><i><font color="Green" title="E74">A96</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_8"/><i><font color="Green" title="E75">A97</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_8"><i><font color="Green" title="E74">A96</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_8"><i><font color="Green" title="E74">A96</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_8"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c24">n1</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_8"><i><font color="Green" title="E74">A96</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_1_8"/><b>then </b>
( <font color="Maroon" title="c24">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_8"><i><font color="Green" title="E74">A96</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_8"><i><font color="Green" title="E75">A97</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_1_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E74">A98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_9"/><i><font color="Green" title="E75">A99</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_9"/><b>then </b><i><font color="Green" title="E76">A100</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>] &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E75">A99</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_9"/>
<font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_9"/><b>then </b><i><font color="Green" title="E77">A101</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_9"/><b>then </b><i><font color="Green" title="E78">A102</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E38:2_2_1_2"><i><font color="Green" title="E42">A54</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_9"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E68">A87</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E75">A99</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E77">A101</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_9"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_1_1_1_1_9"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E75">A99</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E77">A101</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_9"/>
<font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_1_9"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E68">A87</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E75">A99</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E76">A100</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E77">A101</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_9"><i><font color="Green" title="E78">A102</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E75">A103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_10"/><i><font color="Green" title="E76">A104</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_10"/><b>then </b><i><font color="Green" title="E77">A105</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>] &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>] &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E76">A104</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c27">n1</font>;<br/><b>set </b><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c28">m1</font>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_10"/>
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E77">A105</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_10"/><b>then </b><i><font color="Green" title="E78">A106</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c27">n1</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c27">n1</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c28">m1</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c28">m1</font> &amp; <font color="Maroon" title="c28">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E76">A104</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E77">A105</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_10"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c27">n1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c28">m1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E77">A105</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_10"/>
<font color="Maroon" title="c26">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E76">A104</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E77">A105</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_10"><i><font color="Green" title="E78">A106</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E26:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E76">A107</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_11"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_11"/><i><font color="Green" title="E77">A108</font></i>: 
( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_11"><i><font color="Green" title="E77">A108</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_1_11"/><i><font color="Green" title="E78">A109</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_11"><i><font color="Green" title="E77">A108</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c25">n1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_11"><i><font color="Green" title="E77">A108</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_11"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_11"><i><font color="Green" title="E77">A108</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_11"><i><font color="Green" title="E78">A109</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:2_2_1_2_7_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_12"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_12"/>
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_1_12"/><b>then </b><i><font color="Green" title="E77">A110</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_12_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_12_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E73">A95</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_12"><i><font color="Green" title="E77">A110</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_12_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_12_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2"><i><font color="Green" title="E34">A44</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_12"><i><font color="Green" title="E77">A110</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E73">A95</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_12"><i><font color="Green" title="E77">A110</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI: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="E5:2_2_1_2_7_1_1_1_1_1_12"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E28:2_2_1_2_7_1_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E29:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E72">A94</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E30:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E77">A111</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E78">A112</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E79">A113</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E80">A114</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E81">A115</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E82">A116</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E80">A114</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E83">A117</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E80">A114</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E84">A118</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_1_13"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E78">A112</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E85">A119</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">u1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E78">A112</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E81">A115</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E82">A116</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E15:2_2_1_2_7_1_1_1_1_1_13"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E16:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E86">A120</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E78">A112</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E82">A116</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E83">A117</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E84">A118</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E17:2_2_1_2_7_1_1_1_1_1_13"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E18:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E87">A121</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E79">A113</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E81">A115</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E82">A116</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E83">A117</font></i></a></i>;<br/>
<a NAME="E19:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E88">A122</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:2_2_1_2_7_1_1_1_1_1_13"/><i><font color="Green" title="E89">A123</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E82">A116</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E87">A121</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E21:2_2_1_2_7_1_1_1_1_1_13"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E81">A115</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E83">A117</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E88">A122</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E22:2_2_1_2_7_1_1_1_1_1_13"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E81">A115</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E82">A116</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E83">A117</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E85">A119</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E86">A120</font></i></a>, <a class="txt" href="goboard3.html#E20:2_2_1_2_7_1_1_1_1_1_13"><i><font color="Green" title="E89">A123</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E31:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E78">A124</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_1_14"/><b>then </b><i><font color="Green" title="E79">A125</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E80">A126</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E81">A127</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_1_14"/><b>then </b><i><font color="Green" title="E82">A128</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E83">A129</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E84">A130</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_1_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_1_14"/><b>then </b><i><font color="Green" title="E85">A131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E10:2_2_1_2_7_1_1_1_1_1_14"/><i><font color="Green" title="E86">A132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_1"/><i><font color="Green" title="E87">A133</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_14_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_1"><i><font color="Green" title="E87">A133</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2"/><b>then </b><i><font color="Green" title="E87">A134</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_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="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E88">A135</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"/><i><font color="Green" title="E89">A136</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1_1"><i><font color="Green" title="E89">A136</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E89">A137</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A138</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A139</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E81">A127</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E92">A140</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><i><font color="Green" title="E93">A141</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E91">A139</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E94">A142</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E79">A125</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E95">A143</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E83">A129</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A135</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E94">A142</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A144</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E97">A145</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font> &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E71">A90</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E80">A126</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E86">A132</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E93">A141</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E95">A143</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E96">A144</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E97">A145</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A146</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E89">A137</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E90">A138</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E92">A140</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E93">A141</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E19:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E80">A126</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E82">A128</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E86">A132</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/><a NAME="E20:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E22:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E98">A146</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14_1_2"><i><font color="Green" title="E87">A134</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A147</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_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="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E89">A148</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E90">A149</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E82">A128</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A150</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E92">A151</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E93">A152</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A148</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A153</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A152</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E95">A154</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E96">A155</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E97">A156</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A154</font></i></a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E98">A157</font></i>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A153</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A156</font></i></a></i>;<br/><a NAME="E19:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">A158</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E20:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E100">A159</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A157</font></i></a></i>;<br/><a NAME="E21:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E101">A160</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c25">u1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E82">A128</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A149</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A150</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E22:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E102">A161</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E80">A126</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E86">A132</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A150</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E92">A151</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A156</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E23:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E69">A88</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A156</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A158</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>hence </b><a NAME="E24:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A150</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A155</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A156</font></i></a>, <a class="txt" href="goboard3.html#E20:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E100">A159</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E101">A160</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E102">A161</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E89">A162</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E88">A147</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E90">A163</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E82">A128</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A162</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E91">A164</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E81">A127</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A162</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E92">A165</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A162</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E93">A166</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E94">A167</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A168</font></i>: 
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E82">A128</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E84">A130</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E96">A169</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E79">A125</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E91">A164</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E93">A166</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E97">A170</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E93">A166</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E94">A167</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E95">A168</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E92">A165</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E14:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E30:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E77">A111</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_14"><i><font color="Green" title="E85">A131</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E90">A163</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E96">A169</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E97">A170</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_14_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_14_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E32:2_2_1_2_7_1_1_1_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E78">A171</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E79">A172</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>and </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E80">A173</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E4:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E81">A174</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E79">A172</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c29">lp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c30">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_15"/>
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_1_15"/><b>then </b><i><font color="Green" title="E82">A175</font></i>: 
( <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E79">A172</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E83">A176</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c26">p2</font>,<font color="Maroon" title="c25">p1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E81">A174</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E82">A175</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<i><font color="Green" title="E84">A177</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c28">q2</font>,<font color="Maroon" title="c27">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E81">A174</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E82">A175</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E85">A178</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E80">A173</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_1_15"/><b>then </b><i><font color="Green" title="E86">A179</font></i>: 
 ex <font color="Olive" title="b1">tn</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E83">A176</font></i></a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_1_15"/><i><font color="Green" title="E87">A180</font></i>: 
 ex <font color="Olive" title="b1">tm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E84">A177</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E85">A178</font></i></a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_1_1_1_1_15"/>
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E78">A171</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E79">A172</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:2_2_1_2_7_1_1_1_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E86">A179</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_15"><i><font color="Green" title="E87">A180</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E78">A181</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E34:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E79">A182</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_16"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_16"/><i><font color="Green" title="E80">A183</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font> &amp; <font color="Maroon" title="c25">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_16"/><i><font color="Green" title="E81">A184</font></i>: 
( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E80">A183</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_16"/>
( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E81">A184</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_1_16"/><b>then </b><i><font color="Green" title="E82">A185</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_1_16"/><i><font color="Green" title="E83">A186</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E71">A90</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E76">A107</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E82">A185</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E10:2_2_1_2_7_1_1_1_1_1_16"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E81">A184</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E71">A90</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E80">A183</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E81">A184</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_16"><i><font color="Green" title="E83">A186</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E80">A187</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_17"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_17"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_17"/><i><font color="Green" title="E81">A188</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">X</font> &amp; <font color="Maroon" title="c24">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_17"/><i><font color="Green" title="E82">A189</font></i>: 
( <font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E81">A188</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_17"/><i><font color="Green" title="E83">A190</font></i>: 
( <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E82">A189</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_1_17"/><i><font color="Green" title="E84">A191</font></i>: 
( <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E83">A190</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q2</font>,<font color="Maroon" title="c26">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E82">A189</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E84">A191</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<a NAME="E10:2_2_1_2_7_1_1_1_1_1_17"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E81">A188</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E82">A189</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_1_17"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_17"><i><font color="Green" title="E83">A190</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E81">A192</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_18"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_18_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_18_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_18_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_18_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_18_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E82">A193</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E79">A182</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_18_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_18_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_18_1_2_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_18_1_2"><i><font color="Green" title="E82">A193</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_18_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E35:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E80">A187</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_18_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E37:2_2_1_2_7_1_1_1_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E82">A194</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E83">A195</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E84">A196</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E83">A195</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">l1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c26">l2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_19"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_1_19"/><b>then </b><i><font color="Green" title="E85">A197</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_1_19"/><b>then </b><i><font color="Green" title="E86">A198</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">ql</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E87">A199</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_1_19"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E88">A200</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E89">A201</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E85">A197</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E90">A202</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">qq</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E85">A197</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E91">A203</font></i>: 
( <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E86">A198</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E92">A204</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">qq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E87">A199</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E88">A200</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E89">A201</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E90">A202</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E69">A88</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E91">A203</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<a NAME="E15:2_2_1_2_7_1_1_1_1_1_19"/><i><font color="Green" title="E93">A205</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1"/>
( <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/><i><font color="Green" title="E94">A206</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/><b>then </b><i><font color="Green" title="E95">A207</font></i>: 
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E93">A205</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_1"><i><font color="Green" title="E94">A206</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19_3_1_1"><i><font color="Green" title="E95">A207</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_19_3_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2"/><b>then </b><i><font color="Green" title="E94">A208</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c24">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_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="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1"/>
( <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><i><font color="Green" title="E95">A209</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"/><i><font color="Green" title="E96">A210</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_1"><i><font color="Green" title="E96">A210</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A211</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A212</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A213</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">A214</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"/><i><font color="Green" title="E100">A215</font></i>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E92">A204</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E91">A203</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A215</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A215</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A216</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E96">A211</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E99">A214</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c30">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E101">A217</font></i>: 
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E100">A216</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E102">A218</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A209</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E97">A212</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E98">A213</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"><i><font color="Green" title="E101">A217</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1_3"><i><font color="Green" title="E102">A218</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A219</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2"><i><font color="Green" title="E94">A208</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E96">A220</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E95">A219</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/>
( <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E95">A219</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E97">A221</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A220</font></i></a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E98">A222</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_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="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A223</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A223</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A223</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E81">A192</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E98">A222</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E99">A224</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E100">A225</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E101">A226</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_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="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E102">A227</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A228</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A229</font></i>: 
 ex <font color="Olive" title="b1">qx</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E92">A204</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E102">A227</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E105">A230</font></i>: 
( <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E97">A221</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c34">q1l</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E106">A231</font></i>: 
( <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A230</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A225</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E107">A232</font></i>: 
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c33">q2</font>,<font color="Maroon" title="c32">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E97">A221</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E106">A231</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A233</font></i>: 
 ex <font color="Olive" title="b1">qy</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A220</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E103">A228</font></i></a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E75">A103</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E86">A198</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A230</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E107">A232</font></i></a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E104">A229</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E108">A233</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E99">A224</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E82">A194</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_19"><i><font color="Green" title="E84">A196</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A225</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E101">A226</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E78">A181</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A220</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_1_19_3_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_19_3_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:2_2_1_2_7_1_1_1_1_1_19"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E38:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E39:2_2_1_2_7_1_1_1_1_1"/><i><font color="Green" title="E82">A234</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>set </b><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<b>set </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_20"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_1_20"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_1_20"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_21"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_2_1_2_7_1_1_1_1_1_21"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_21"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_21"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_21"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E41:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E20:2_2_1_2"><i><font color="Green" title="E28">A32</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E82">A234</font></i></a>, <a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></i>;<br/><b>thus </b><a NAME="E42:2_2_1_2_7_1_1_1_1_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c24">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:2_2_1_2_7_1_1_1_1_1_22"/><i><font color="Green" title="E83">A235</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green" title="E84">A236</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_1"/><i><font color="Green" title="E85">A237</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>let </b><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_1"/><i><font color="Green" title="E86">A238</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/><i><font color="Green" title="E87">A239</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E88">A240</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E86">A238</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><i><font color="Green" title="E87">A239</font></i></a></i>;<br/><b>thus </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E68">A87</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E86">A238</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><i><font color="Green" title="E87">A239</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><i><font color="Green" title="E88">A240</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E67">A86</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E86">A238</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><i><font color="Green" title="E87">A239</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_1_1_1_1"><i><font color="Green" title="E88">A240</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E85">A237</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/><b>then </b><i><font color="Green" title="E87">A241</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font> &amp; <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E83">A235</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E85">A237</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
<font color="Maroon" title="c26">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j</font>
 ;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/><b>then </b><i><font color="Green" title="E88">A242</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"><i><font color="Green" title="E87">A241</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"><i><font color="Green" title="E87">A241</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E74">A98</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_1"><i><font color="Green" title="E86">A238</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_1_1_1_2"><i><font color="Green" title="E88">A242</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_1"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_2"/><i><font color="Green" title="E85">A243</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font> &amp; <font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2"/><i><font color="Green" title="E86">A244</font></i>: 
( <font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E85">A243</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><i><font color="Green" title="E87">A245</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_2_1_1"><i><font color="Green" title="E87">A245</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E85">A243</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="topreal3.html#T27" title="TOPREAL3:th.27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><i><font color="Green" title="E87">A246</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><b>then </b><i><font color="Green" title="E88">A247</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><i><font color="Green" title="E89">A248</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E87">A246</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><i><font color="Green" title="E90">A249</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E84">A236</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E88">A247</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E89">A248</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><b>then </b><i><font color="Green" title="E91">A250</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/><i><font color="Green" title="E92">A251</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_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="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A250</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A251</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><i><font color="Green" title="E93">A252</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E85">A243</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A249</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><i><font color="Green" title="E93">A252</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A249</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><i><font color="Green" title="E93">A252</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A250</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A251</font></i></a>, <a class="ref" href="topreal1.html#T7" title="TOPREAL1:th.7">TOPREAL1:7</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E85">A243</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A249</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A250</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A251</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><i><font color="Green" title="E93">A253</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E85">A243</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2"><i><font color="Green" title="E86">A244</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A249</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><i><font color="Green" title="E93">A253</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A249</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><i><font color="Green" title="E93">A253</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:2_2_1_2_7_1_1_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E5:2_2_1_2_7_1_1_1_1_1_22"/><b>then </b><i><font color="Green" title="E85">A254</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="goboard2.html#T8" title="GOBOARD2:th.8">GOBOARD2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E85">A254</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T11" title="GOBOARD2:th.11">GOBOARD2:11</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_2"/><i><font color="Green" title="E86">A255</font></i>: 
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E70">A89</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) );<br/><div><i><font color="Green" title="E87">A256</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E83">A235</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"/><i><font color="Green" title="E88">A257</font></i>: 
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"><i><font color="Green" title="E88">A257</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E86">A255</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:2_2_1_2_7_1_1_1_1_1_22_3_2"/><i><font color="Green" title="E88">A258</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><b>consider </b><font color="Maroon" title="c27">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_3_2"/><i><font color="Green" title="E89">A259</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E88">A258</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E87">A256</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E90">A260</font></i>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E91">A261</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> &amp; <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E92">A262</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E93">A263</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E83">A235</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A260</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E92">A262</font></i></a></i>
;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E94">A264</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E86">A255</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E95">A265</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E83">A235</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A260</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E91">A261</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E96">A266</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E97">A267</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E96">A266</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E98">A268</font></i>: 
( <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E84">A236</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E95">A265</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E96">A266</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E99">A269</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>,<font color="Maroon" title="c30">q</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A260</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E93">A263</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E97">A267</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E69">A88</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E98">A268</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E99">A269</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E86">A255</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A260</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E94">A264</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E97">A267</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E98">A268</font></i></a></i>;<br/><b>hence </b><a NAME="E19:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E86">A255</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E90">A270</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp; <font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E91">A271</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E92">A272</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a></i>;<br/><div><i><font color="Green" title="E93">A273</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A270</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A271</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E94">A274</font></i>: 
( <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_1_22"><i><font color="Green" title="E84">A236</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E89">A259</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A270</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A271</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E92">A272</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lma</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E95">A275</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c30">qa1</font>,<font color="Maroon" title="c29">qa</font>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A270</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A271</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E94">A274</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E95">A275</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2"><i><font color="Green" title="E86">A255</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A270</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A271</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E92">A272</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E93">A273</font></i></a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_1_22_3_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E43:2_2_1_2_7_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E44:2_2_1_2_7_1_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E22:2_2_1_2"><i><font color="Green" title="E30">A34</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E46:2_2_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E47:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E83">A276</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E48:2_2_1_2_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E84">A277</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>thus </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E84">A277</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c23">m1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E84">A277</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a></i>
;<br/><b>thus </b><a NAME="E51:2_2_1_2_7_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E83">A276</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_1_1_1_1"><i><font color="Green" title="E84">A277</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_2"/><i><font color="Green" title="E60">A278</font></i>: 
<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E30:2_2_1_2"><i><font color="Green" title="E36">A48</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_2"><i><font color="Green" title="E60">A278</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E60">A279</font></i>: 
<font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c10">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E61">A280</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">GOBOARD2:sch 1</a>();<br/></i><b>take </b><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>;<br/><div><i><font color="Green" title="E62">A281</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_1"/>
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E63">A282</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_1"><i><font color="Green" title="E63">A282</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_1"/>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E6:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E63">A283</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_2"/><i><font color="Green" title="E64">A284</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><b>take </b><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c7">i1</font>;<br/><b>take </b><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_2"><i><font color="Green" title="E64">A284</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E64">A285</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E65">A286</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_3"/>
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_3"/><b>then </b><i><font color="Green" title="E66">A287</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c23">l1</font>, <font color="Maroon" title="c24">l2</font>, <font color="Maroon" title="c25">l3</font>, <font color="Maroon" title="c26">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font> )
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_3"/><b>then </b>
( <font color="Maroon" title="c23">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c24">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c25">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c26">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_3"><i><font color="Green" title="E66">A287</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">l1</font>, <font color="Maroon" title="c23">l2</font>, <font color="Maroon" title="c24">l3</font>, <font color="Maroon" title="c25">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_4"/><i><font color="Green" title="E66">A288</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_4"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_4"/><b>then </b>
( <font color="Maroon" title="c22">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c23">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c24">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c25">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_4"><i><font color="Green" title="E66">A288</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l4</font></span>)</span></span>)</span> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E11:2_2_1_2_7_1_1_1_1_3"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E65">A286</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E64">A285</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon" title="c22">lk</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_3"/>
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E14:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E66">A289</font></i>: 
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E15:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E67">A290</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E68">A291</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E38:2_2_1_2"><i><font color="Green" title="E42">A54</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green" title="E69">A292</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E68">A291</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E70">A293</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_6"/><i><font color="Green" title="E71">A294</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_6"/><b>then </b><i><font color="Green" title="E72">A295</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a></i>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_6"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_6"/><b>then </b>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_6"><i><font color="Green" title="E72">A295</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_6"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_6"><i><font color="Green" title="E71">A294</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E20:2_2_1_2_7_1_1_1_1_3"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E21:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E71">A296</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/><a NAME="E22:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E72">A297</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_7"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_7"/><i><font color="Green" title="E73">A298</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_7"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_3_7"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_7"><i><font color="Green" title="E73">A298</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_7"/><b>then </b>
( <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_7"><i><font color="Green" title="E73">A298</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_7"/>
contradiction
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E73">A299</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_8"/><i><font color="Green" title="E74">A300</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_8"/><b>then </b><i><font color="Green" title="E75">A301</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_8"/><b>then </b><i><font color="Green" title="E76">A302</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_8"/><b>then </b><i><font color="Green" title="E77">A303</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i2</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E38:2_2_1_2"><i><font color="Green" title="E42">A54</font></i></a>, <a class="txt" href="goboard3.html#E42:2_2_1_2"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E45:2_2_1_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_8"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E67">A290</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E74">A300</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_8"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E74">A300</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E76">A302</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c8">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E67">A290</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E74">A300</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E75">A301</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_8"><i><font color="Green" title="E77">A303</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E74">A304</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_9"/><i><font color="Green" title="E75">A305</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_9"/><b>then </b><i><font color="Green" title="E76">A306</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E62">A281</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E63">A283</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><b>set </b><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font>;<br/><b>set </b><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_9"/>
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E47:2_2_1_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E76">A306</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_9"/><b>then </b><i><font color="Green" title="E77">A307</font></i>: 
( <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c11">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E75">A305</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E76">A306</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_9"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E76">A306</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_9"/>
<font color="Maroon" title="c25">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E75">A305</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E76">A306</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_9"><i><font color="Green" title="E77">A307</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E25:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E75">A308</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_10"/><i><font color="Green" title="E76">A309</font></i>: 
( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_10"/><i><font color="Green" title="E77">A310</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_10"><i><font color="Green" title="E76">A309</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_3_10"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_10"><i><font color="Green" title="E76">A309</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_10"><i><font color="Green" title="E77">A310</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_10"><i><font color="Green" title="E76">A309</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_10"><i><font color="Green" title="E77">A310</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:2_2_1_2_7_1_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_11"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_11"/>
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_3_11"/><b>then </b><i><font color="Green" title="E76">A311</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_11_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_11_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_11_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E72">A297</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_11"><i><font color="Green" title="E76">A311</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_11_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_11_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_11_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2"><i><font color="Green" title="E34">A44</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_11"><i><font color="Green" title="E76">A311</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E72">A297</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_11"><i><font color="Green" title="E76">A311</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI: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="E5:2_2_1_2_7_1_1_1_1_3_11"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E27:2_2_1_2_7_1_1_1_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E28:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E71">A296</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E29:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E76">A312</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E77">A313</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b><i><font color="Green" title="E78">A314</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b><i><font color="Green" title="E79">A315</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E80">A316</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E81">A317</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E79">A315</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E82">A318</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E79">A315</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E83">A319</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E84">A320</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_3_12"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E77">A313</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b><i><font color="Green" title="E85">A321</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c26">u2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E77">A313</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E80">A316</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E81">A317</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E82">A318</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E83">A319</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E84">A320</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E16:2_2_1_2_7_1_1_1_1_3_12"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b><i><font color="Green" title="E86">A322</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E78">A314</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E80">A316</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E81">A317</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E82">A318</font></i></a></i>;<br/>
<a NAME="E18:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b><i><font color="Green" title="E87">A323</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:2_2_1_2_7_1_1_1_1_3_12"/><i><font color="Green" title="E88">A324</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E81">A317</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E86">A322</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E20:2_2_1_2_7_1_1_1_1_3_12"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E80">A316</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E82">A318</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E87">A323</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E21:2_2_1_2_7_1_1_1_1_3_12"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E81">A317</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E85">A321</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_3_12"><i><font color="Green" title="E88">A324</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E30:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13"/><i><font color="Green" title="E77">A325</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_3_13"/><b>then </b><i><font color="Green" title="E78">A326</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_3_13"/><i><font color="Green" title="E79">A327</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_13"/><i><font color="Green" title="E80">A328</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_3_13"/><b>then </b><i><font color="Green" title="E81">A329</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_13"/><i><font color="Green" title="E82">A330</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_3_13"/><i><font color="Green" title="E83">A331</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_3_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_3_13"/><b>then </b><i><font color="Green" title="E84">A332</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E83">A331</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_1"/><i><font color="Green" title="E85">A333</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_13_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_1"><i><font color="Green" title="E85">A333</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2"/><b>then </b><i><font color="Green" title="E85">A334</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_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="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><i><font color="Green" title="E86">A335</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"/><i><font color="Green" title="E87">A336</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E86">A335</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1_1"><i><font color="Green" title="E87">A336</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E87">A337</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E88">A338</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E89">A339</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E80">A328</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E86">A335</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A340</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><i><font color="Green" title="E91">A341</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E86">A335</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E89">A339</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><i><font color="Green" title="E92">A342</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E78">A326</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E86">A335</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A343</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E82">A330</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E83">A331</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E86">A335</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E83">A331</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E92">A342</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A344</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><i><font color="Green" title="E95">A345</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font> &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E70">A293</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E79">A327</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E82">A330</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E91">A341</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E93">A343</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E94">A344</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E95">A345</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A346</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A337</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E88">A338</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E90">A340</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E91">A341</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E19:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E79">A327</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E81">A329</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E82">A330</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/><a NAME="E20:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E22:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E96">A346</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13_1_2"><i><font color="Green" title="E85">A334</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E86">A347</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_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="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E87">A348</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E88">A349</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E87">A348</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E89">A350</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E83">A331</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E87">A348</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A351</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E87">A348</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E91">A352</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E87">A348</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A350</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A350</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E92">A353</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A350</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A354</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E94">A355</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E95">A356</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A354</font></i></a></i>;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E96">A357</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E92">A353</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A355</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A356</font></i></a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A358</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A355</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E19:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E98">A359</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A355</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A357</font></i></a></i>;<br/><a NAME="E20:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A360</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E77">A325</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E79">A327</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E81">A329</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E82">A330</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A349</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A351</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A352</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A355</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A356</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c26">u2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E68">A291</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A356</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A358</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A359</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>hence </b><a NAME="E22:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A351</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A355</font></i></a>, <a class="txt" href="goboard3.html#E20:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A360</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E87">A361</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"><i><font color="Green" title="E86">A347</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A362</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E81">A329</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E87">A361</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E89">A363</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E80">A328</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E87">A361</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E90">A364</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E87">A361</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E91">A365</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E92">A366</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E81">A329</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E83">A331</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E91">A365</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E78">A326</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A363</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E90">A364</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E91">A365</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E92">A366</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E29:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E76">A312</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_13"><i><font color="Green" title="E84">A332</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A362</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_13_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_13_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E31:2_2_1_2_7_1_1_1_1_3"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E77">A367</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E78">A368</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>and </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E79">A369</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E4:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E80">A370</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E78">A368</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c29">lp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c30">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_14"/>
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_3_14"/><b>then </b><i><font color="Green" title="E81">A371</font></i>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E78">A368</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E82">A372</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">p1</font>,<font color="Maroon" title="c26">p2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E80">A370</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E81">A371</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<i><font color="Green" title="E83">A373</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E80">A370</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E81">A371</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E84">A374</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E79">A369</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_3_14"/><b>then </b><i><font color="Green" title="E85">A375</font></i>: 
 ex <font color="Olive" title="b1">tn</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E82">A372</font></i></a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_3_14"/><i><font color="Green" title="E86">A376</font></i>: 
 ex <font color="Olive" title="b1">tm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E83">A373</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E84">A374</font></i></a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_1_1_1_3_14"/>
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E77">A367</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E78">A368</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:2_2_1_2_7_1_1_1_1_3_14"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E85">A375</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_3_14"><i><font color="Green" title="E86">A376</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E77">A377</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E33:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E78">A378</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_15"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_15"/><i><font color="Green" title="E79">A379</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font> &amp; <font color="Maroon" title="c25">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_15"/><i><font color="Green" title="E80">A380</font></i>: 
( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E79">A379</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_15"/>
( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E80">A380</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_1_1_1_3_15"/><b>then </b><i><font color="Green" title="E81">A381</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_3_15"/><i><font color="Green" title="E82">A382</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E70">A293</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E75">A308</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E81">A381</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E10:2_2_1_2_7_1_1_1_1_3_15"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E80">A380</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E70">A293</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E79">A379</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E80">A380</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_15"><i><font color="Green" title="E82">A382</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E34:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E79">A383</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_16"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_16"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_16"/><i><font color="Green" title="E80">A384</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">X</font> &amp; <font color="Maroon" title="c24">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_16"/><i><font color="Green" title="E81">A385</font></i>: 
( <font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E80">A384</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_16"/><i><font color="Green" title="E82">A386</font></i>: 
( <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E81">A385</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_3_16"/><i><font color="Green" title="E83">A387</font></i>: 
( <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E82">A386</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c27">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E81">A385</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E83">A387</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<a NAME="E10:2_2_1_2_7_1_1_1_1_3_16"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E80">A384</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E81">A385</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_1_1_1_3_16"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_16"><i><font color="Green" title="E82">A386</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E80">A388</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_17"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_17_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_17_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_17_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_17_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_17_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_17_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_17_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_17_1_2"/><b>then </b><i><font color="Green" title="E81">A389</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E78">A378</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_17_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_17_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_17_1_2_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_17_1_2"><i><font color="Green" title="E81">A389</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_17_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E79">A383</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_17_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E36:2_2_1_2_7_1_1_1_1_3"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E81">A390</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E82">A391</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E83">A392</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E82">A391</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">l1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c26">l2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_18"/>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_1_1_1_3_18"/><b>then </b><i><font color="Green" title="E84">A393</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_1_1_1_3_18"/><b>then </b><i><font color="Green" title="E85">A394</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">ql</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E86">A395</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_2_1_2_7_1_1_1_1_3_18"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E87">A396</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<a NAME="E11:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E88">A397</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E84">A393</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E89">A398</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">qq</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E84">A393</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E90">A399</font></i>: 
( <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E85">A394</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E91">A400</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c27">qq</font>
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E86">A395</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E87">A396</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E88">A397</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E89">A398</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E68">A291</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E90">A399</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/>
<a NAME="E15:2_2_1_2_7_1_1_1_1_3_18"/><i><font color="Green" title="E92">A401</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1"/>
( <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/><i><font color="Green" title="E93">A402</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/><b>then </b><i><font color="Green" title="E94">A403</font></i>: 
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E92">A401</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_1"><i><font color="Green" title="E93">A402</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18_3_1_1"><i><font color="Green" title="E94">A403</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_18_3_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2"/><b>then </b><i><font color="Green" title="E93">A404</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c24">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_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="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1"/>
( <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><i><font color="Green" title="E94">A405</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"/><i><font color="Green" title="E95">A406</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_1"><i><font color="Green" title="E95">A406</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E95">A407</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A408</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A409</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A410</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"/><i><font color="Green" title="E99">A411</font></i>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E91">A400</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E90">A399</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"><i><font color="Green" title="E99">A411</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_2"><i><font color="Green" title="E99">A411</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">A412</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A407</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E98">A410</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c30">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E100">A413</font></i>: 
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E99">A412</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E101">A414</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E94">A405</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E96">A408</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E97">A409</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"><i><font color="Green" title="E100">A413</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1_3"><i><font color="Green" title="E101">A414</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E94">A415</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2"><i><font color="Green" title="E93">A404</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A416</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E94">A415</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/>
( <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E94">A415</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E96">A417</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E95">A416</font></i></a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E97">A418</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_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="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E98">A419</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A419</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A419</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E35:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E80">A388</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E97">A418</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E98">A420</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E99">A421</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E100">A422</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_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="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E101">A423</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E102">A424</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A425</font></i>: 
 ex <font color="Olive" title="b1">qx</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E91">A400</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E101">A423</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E104">A426</font></i>: 
( <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E96">A417</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c34">q1l</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E105">A427</font></i>: 
( <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E104">A426</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E99">A421</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E106">A428</font></i>: 
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c32">q1</font>,<font color="Maroon" title="c33">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E96">A417</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A427</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E107">A429</font></i>: 
 ex <font color="Olive" title="b1">qy</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E95">A416</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E102">A424</font></i></a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E74">A304</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E85">A394</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E104">A426</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E106">A428</font></i></a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E103">A425</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E107">A429</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E98">A420</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E81">A390</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_18"><i><font color="Green" title="E83">A392</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E99">A421</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A422</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E32:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E77">A377</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E95">A416</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_3_18_3_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_18_3_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:2_2_1_2_7_1_1_1_1_3_18"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E37:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E38:2_2_1_2_7_1_1_1_1_3"/><i><font color="Green" title="E81">A430</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>set </b><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<b>set </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_19"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_1_1_1_3_19"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_1_1_1_3_19"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_19"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_20"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_2_1_2_7_1_1_1_1_3_20"/>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_20"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_20"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_20"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E40:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E20:2_2_1_2"><i><font color="Green" title="E28">A32</font></i></a>, <a class="txt" href="goboard3.html#E38:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E81">A430</font></i></a>, <a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></i>;<br/><b>thus </b><a NAME="E41:2_2_1_2_7_1_1_1_1_3"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c24">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:2_2_1_2_7_1_1_1_1_3_21"/><i><font color="Green" title="E82">A431</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green" title="E83">A432</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_1"/><i><font color="Green" title="E84">A433</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>let </b><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_1"/><i><font color="Green" title="E85">A434</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/><i><font color="Green" title="E86">A435</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E87">A436</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E85">A434</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_1_1_1"><i><font color="Green" title="E86">A435</font></i></a></i>;<br/><b>thus </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2"><i><font color="Green" title="E43">A55</font></i></a>, <a class="txt" href="goboard3.html#E43:2_2_1_2"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="goboard3.html#E44:2_2_1_2"><i><font color="Green" title="E48">A60</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E67">A290</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E85">A434</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_1_1_1"><i><font color="Green" title="E86">A435</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_1_1_1"><i><font color="Green" title="E87">A436</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">l1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_3_21_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E66">A289</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E85">A434</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_1_1_1"><i><font color="Green" title="E86">A435</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_1_1_1"><i><font color="Green" title="E87">A436</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/><b>then </b><i><font color="Green" title="E86">A437</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E84">A433</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/><i><font color="Green" title="E87">A438</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21_1_1_2"><i><font color="Green" title="E86">A437</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/><b>then </b><i><font color="Green" title="E88">A439</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font> &amp; <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E82">A431</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E84">A433</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/><b>then </b><i><font color="Green" title="E89">A440</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_1_1_2"><i><font color="Green" title="E87">A438</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/>
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_1_1_2"><i><font color="Green" title="E88">A439</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_3_21_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">l1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E73">A299</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_1"><i><font color="Green" title="E85">A434</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_1_1_2"><i><font color="Green" title="E89">A440</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_2"/><i><font color="Green" title="E84">A441</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font> &amp; <font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2"/><i><font color="Green" title="E85">A442</font></i>: 
( <font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E84">A441</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><i><font color="Green" title="E86">A443</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_2_1_1"><i><font color="Green" title="E86">A443</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E84">A441</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="topreal3.html#T27" title="TOPREAL3:th.27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_1_1_1_3_21_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><i><font color="Green" title="E86">A444</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><b>then </b><i><font color="Green" title="E87">A445</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><i><font color="Green" title="E88">A446</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E86">A444</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><i><font color="Green" title="E89">A447</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E83">A432</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E87">A445</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E88">A446</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><b>then </b><i><font color="Green" title="E90">A448</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/><i><font color="Green" title="E91">A449</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_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="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E90">A448</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E91">A449</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><i><font color="Green" title="E92">A450</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E84">A441</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E89">A447</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><i><font color="Green" title="E92">A450</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E89">A447</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><i><font color="Green" title="E92">A450</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E90">A448</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E91">A449</font></i></a>, <a class="ref" href="topreal1.html#T7" title="TOPREAL1:th.7">TOPREAL1:7</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E84">A441</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E89">A447</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E90">A448</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E91">A449</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><i><font color="Green" title="E92">A451</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E84">A441</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2"><i><font color="Green" title="E85">A442</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E89">A447</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><i><font color="Green" title="E92">A451</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_2_1_2"><i><font color="Green" title="E89">A447</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><i><font color="Green" title="E92">A451</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:2_2_1_2_7_1_1_1_1_3_21_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E5:2_2_1_2_7_1_1_1_1_3_21"/><b>then </b><i><font color="Green" title="E84">A452</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="goboard2.html#T8" title="GOBOARD2:th.8">GOBOARD2:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E84">A452</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T11" title="GOBOARD2:th.11">GOBOARD2:11</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"/><i><font color="Green" title="E85">A453</font></i>: 
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E69">A292</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) );<br/><div><i><font color="Green" title="E86">A454</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E82">A431</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/><i><font color="Green" title="E87">A455</font></i>: 
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"><i><font color="Green" title="E87">A455</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E85">A453</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:2_2_1_2_7_1_1_1_1_3_21_3_1_2"/><i><font color="Green" title="E87">A456</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><b>consider </b><font color="Maroon" title="c27">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"/><i><font color="Green" title="E88">A457</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E87">A456</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A454</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_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="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E89">A458</font></i>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E90">A459</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E91">A460</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E92">A461</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E82">A431</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E89">A458</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E91">A460</font></i></a></i>
;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E93">A462</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E94">A463</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E82">A431</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E89">A458</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E90">A459</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E95">A464</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E96">A465</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E95">A464</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E97">A466</font></i>: 
( <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E83">A432</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E94">A463</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E95">A464</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E98">A467</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c30">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E89">A458</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E92">A461</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E96">A465</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E17:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c10">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E68">A291</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E97">A466</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E98">A467</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E89">A458</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E93">A462</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E96">A465</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E97">A466</font></i></a></i>;<br/><b>hence </b><a NAME="E19:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E89">A468</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp; <font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E90">A469</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E91">A470</font></i>: 
<font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a></i>;<br/><div><i><font color="Green" title="E92">A471</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/>
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><a NAME="E2:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E89">A468</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A469</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E93">A472</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_1_1_1_3_21"><i><font color="Green" title="E83">A432</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A457</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E89">A468</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A469</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A470</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lma</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> ;<br/><a NAME="E9:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E94">A473</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">qa</font>,<font color="Maroon" title="c30">qa1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E89">A468</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A469</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E93">A472</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E94">A473</font></i></a>, <a class="ref" href="topreal3.html#T15" title="TOPREAL3:th.15">TOPREAL3:15</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2"><i><font color="Green" title="E85">A453</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E89">A468</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A469</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A470</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E92">A471</font></i></a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_1_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_1_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E42:2_2_1_2_7_1_1_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E43:2_2_1_2_7_1_1_1_1_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E22:2_2_1_2"><i><font color="Green" title="E30">A34</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E45:2_2_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E60">A279</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E46:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E82">A474</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E47:2_2_1_2_7_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E83">A475</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E61">A280</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E83">A475</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_1"><i><font color="Green" title="E59">A71</font></i></a></i>
;<br/><br/><b>thus </b><a NAME="E49:2_2_1_2_7_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E46:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E82">A474</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2_7_1_1_1_1_3"><i><font color="Green" title="E83">A475</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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="E3:2_2_1_2_7_1_1"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2"/><i><font color="Green" title="E59">A476</font></i>: 
<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c17">ppi</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>;<br/><b>set </b><font color="Maroon" title="c18">pj</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_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="E1:2_2_1_2_7_1_2_1_1"/>
( <font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">j1</font> or <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> or <font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E60">A477</font></i>: 
<font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><i><font color="Green" title="E61">A478</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E62">A479</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_1"><i><font color="Green" title="E62">A479</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">w</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_1"><i><font color="Green" title="E62">A479</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w</font> &amp; <font color="Maroon" title="c21">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_1_1"/>
( <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c8">i2</font>;<br/><div><i><font color="Green" title="E62">A480</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_2"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c22">p</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">m</font>,<font color="Maroon" title="c8">i2</font>;<br/><b>thus </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">n</font>,<font color="Maroon" title="c22">p</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E63">A481</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_4.html#S1" title="FINSEQ_4:sch.1">FINSEQ_4:sch 1</a>(<a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E62">A480</font></i></a>);<br/></i><a NAME="E7:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E64">A482</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_3"/><i><font color="Green" title="E65">A483</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c23">m</font>;<br/><b>take </b><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c8">i2</font>;<br/><b>thus </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c25">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c25">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_3"><i><font color="Green" title="E65">A483</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E65">A484</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E66">A485</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_4"/><i><font color="Green" title="E67">A486</font></i>: 
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E68">A487</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font>, <font color="Maroon" title="c24">m2</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_4"><i><font color="Green" title="E67">A486</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_4"/><i><font color="Green" title="E69">A488</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">m1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">m2</font>,<font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_4"><i><font color="Green" title="E67">A486</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c25">l1</font>, <font color="Maroon" title="c26">l2</font>, <font color="Maroon" title="c27">l3</font>, <font color="Maroon" title="c28">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l1</font>,<font color="Maroon" title="c26">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">l1</font>,<font color="Maroon" title="c26">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">l4</font> )
 ;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c25">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">m1</font> &amp; <font color="Maroon" title="c26">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c27">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">m2</font> &amp; <font color="Maroon" title="c28">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_4"><i><font color="Green" title="E68">A487</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_4"><i><font color="Green" title="E69">A488</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">l1</font>, <font color="Maroon" title="c23">l2</font>, <font color="Maroon" title="c24">l3</font>, <font color="Maroon" title="c25">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_5"/><i><font color="Green" title="E67">A489</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_5"/><b>then </b><i><font color="Green" title="E68">A490</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_5"><i><font color="Green" title="E67">A489</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_5"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c26">m1</font>,<font color="Maroon" title="c8">i2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_5"><i><font color="Green" title="E67">A489</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_5"/><b>then </b>
( <font color="Maroon" title="c22">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c23">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c24">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font> &amp; <font color="Maroon" title="c25">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_5"><i><font color="Green" title="E67">A489</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_5"><i><font color="Green" title="E68">A490</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:2_2_1_2_7_1_2_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E66">A485</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E65">A484</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon" title="c22">lk</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E14:2_2_1_2_7_1_2_1_1_1"/>
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E67">A491</font></i>: 
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E68">A492</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E69">A493</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E41:2_2_1_2"><i><font color="Green" title="E45">A57</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green" title="E70">A494</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E69">A493</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E71">A495</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_7"/><i><font color="Green" title="E72">A496</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_7"/><b>then </b><i><font color="Green" title="E73">A497</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_7"><i><font color="Green" title="E72">A496</font></i></a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_7"/><i><font color="Green" title="E74">A498</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c26">m1</font>,<font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_7"><i><font color="Green" title="E72">A496</font></i></a></i>;<br/><b>assume </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_7"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_7"><i><font color="Green" title="E73">A497</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_7"><i><font color="Green" title="E74">A498</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_7"><i><font color="Green" title="E72">A496</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E21:2_2_1_2_7_1_2_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E22:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E72">A499</font></i>: 
( <font color="Maroon" title="c20">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/><a NAME="E23:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E73">A500</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_8"/><i><font color="Green" title="E74">A501</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_8"/><i><font color="Green" title="E75">A502</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_8"><i><font color="Green" title="E74">A501</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_8"><i><font color="Green" title="E74">A501</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_8"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">n1</font>,<font color="Maroon" title="c8">i2</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_8"><i><font color="Green" title="E74">A501</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_1_8"/><b>then </b>
( <font color="Maroon" title="c24">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_8"><i><font color="Green" title="E74">A501</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_8"><i><font color="Green" title="E75">A502</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_1_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E74">A503</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_9"/><i><font color="Green" title="E75">A504</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_9"/><b>then </b><i><font color="Green" title="E76">A505</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>] &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E75">A504</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_9"/>
<font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_9"/><b>then </b><i><font color="Green" title="E77">A506</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_9"/><b>then </b><i><font color="Green" title="E78">A507</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E41:2_2_1_2"><i><font color="Green" title="E45">A57</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_9"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E68">A492</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E75">A504</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E77">A506</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><b>thus </b><a NAME="E8:2_2_1_2_7_1_2_1_1_1_9"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E75">A504</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E77">A506</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_9"/>
<font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_1_9"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E68">A492</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E75">A504</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E76">A505</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E77">A506</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_9"><i><font color="Green" title="E78">A507</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E75">A508</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_10"/><i><font color="Green" title="E76">A509</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_10"/><b>then </b><i><font color="Green" title="E77">A510</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">n</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>] &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>] &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E76">A509</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c8">i2</font>;<br/><b>set </b><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c8">i2</font>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_10"/>
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E77">A510</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_10"/><b>then </b><i><font color="Green" title="E78">A511</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c28">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E76">A509</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E77">A510</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_10"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E77">A510</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_10"/>
<font color="Maroon" title="c26">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E76">A509</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E77">A510</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_10"><i><font color="Green" title="E78">A511</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E26:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E76">A512</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_11"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_11"/><i><font color="Green" title="E77">A513</font></i>: 
( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E61">A478</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_11"><i><font color="Green" title="E77">A513</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_1_11"/><i><font color="Green" title="E78">A514</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_11"><i><font color="Green" title="E77">A513</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_11"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_11"><i><font color="Green" title="E77">A513</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_11"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_11"><i><font color="Green" title="E77">A513</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_11"><i><font color="Green" title="E78">A514</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:2_2_1_2_7_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_12"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_12"/>
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_1_12"/><b>then </b><i><font color="Green" title="E77">A515</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_12_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_12_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E73">A500</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_12"><i><font color="Green" title="E77">A515</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_12_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_12_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2"><i><font color="Green" title="E34">A44</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_12"><i><font color="Green" title="E77">A515</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E73">A500</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_12"><i><font color="Green" title="E77">A515</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI: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="E5:2_2_1_2_7_1_2_1_1_1_12"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E28:2_2_1_2_7_1_2_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E29:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E72">A499</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E30:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E77">A516</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E78">A517</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b><i><font color="Green" title="E79">A518</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b><i><font color="Green" title="E80">A519</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E81">A520</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E82">A521</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E80">A519</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E83">A522</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E80">A519</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E84">A523</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E85">A524</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_1_13"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E78">A517</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b><i><font color="Green" title="E86">A525</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c26">u2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E78">A517</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E81">A520</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E82">A521</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E83">A522</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E84">A523</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E85">A524</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E16:2_2_1_2_7_1_2_1_1_1_13"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b><i><font color="Green" title="E87">A526</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E79">A518</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E81">A520</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E82">A521</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E83">A522</font></i></a></i>;<br/>
<a NAME="E18:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b><i><font color="Green" title="E88">A527</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:2_2_1_2_7_1_2_1_1_1_13"/><i><font color="Green" title="E89">A528</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E82">A521</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E87">A526</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E20:2_2_1_2_7_1_2_1_1_1_13"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E81">A520</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E83">A522</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E88">A527</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/>
<b>hence </b><a NAME="E21:2_2_1_2_7_1_2_1_1_1_13"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E82">A521</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E86">A525</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1_13"><i><font color="Green" title="E89">A528</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E31:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14"/><i><font color="Green" title="E78">A529</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_1_14"/><b>then </b><i><font color="Green" title="E79">A530</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_1_14"/><i><font color="Green" title="E80">A531</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_14"/><i><font color="Green" title="E81">A532</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_1_14"/><b>then </b><i><font color="Green" title="E82">A533</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_14"/><i><font color="Green" title="E83">A534</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_1_14"/><i><font color="Green" title="E84">A535</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_1_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_1_14"/><b>then </b><i><font color="Green" title="E85">A536</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E84">A535</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_1"/><i><font color="Green" title="E86">A537</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_14_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_1"><i><font color="Green" title="E86">A537</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2"/><b>then </b><i><font color="Green" title="E86">A538</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>thus </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_14_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_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="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E87">A539</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"/><i><font color="Green" title="E88">A540</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1_1"><i><font color="Green" title="E88">A540</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E88">A541</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E89">A542</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A543</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E81">A532</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A544</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><i><font color="Green" title="E92">A545</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E90">A543</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E83">A534</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E84">A535</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A546</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E83">A534</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E79">A530</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E87">A539</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E84">A535</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A547</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E95">A548</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font> &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E71">A495</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E80">A531</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E83">A534</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E92">A545</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E93">A546</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E94">A547</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E95">A548</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A549</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E88">A541</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E89">A542</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E91">A544</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E92">A545</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E20:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E80">A531</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E82">A533</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E83">A534</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E22:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E23:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E96">A549</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14_1_2"><i><font color="Green" title="E86">A538</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E87">A550</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>thus </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_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="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E88">A551</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E89">A552</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A551</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E90">A553</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E84">A535</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A551</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A554</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A551</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E92">A555</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A551</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A553</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A553</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A556</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A553</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A557</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E95">A558</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E96">A559</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A557</font></i></a></i>;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E97">A560</font></i>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A556</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A558</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A559</font></i></a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A561</font></i>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A558</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A562</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A558</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A560</font></i></a></i>;<br/><a NAME="E20:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E100">A563</font></i>: 
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c26">u2</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E78">A529</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E80">A531</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E82">A533</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E83">A534</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A552</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A554</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E92">A555</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A558</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A559</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E69">A493</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A559</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A561</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>hence </b><a NAME="E22:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A554</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A558</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A562</font></i></a>, <a class="txt" href="goboard3.html#E20:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E100">A563</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A564</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E87">A550</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E89">A565</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E82">A533</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A564</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E90">A566</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E81">A532</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A564</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E91">A567</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A564</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E92">A568</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E93">A569</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E82">A533</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E84">A535</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E79">A530</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E90">A566</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E91">A567</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E92">A568</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E93">A569</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E30:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E77">A516</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_14"><i><font color="Green" title="E85">A536</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A565</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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

</div><b>end;</b></div><a NAME="E32:2_2_1_2_7_1_2_1_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E78">A570</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E79">A571</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>and </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E80">A572</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E4:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E81">A573</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E79">A571</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c29">lp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c30">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_15"/>
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_1_15"/><b>then </b><i><font color="Green" title="E82">A574</font></i>: 
( <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E79">A571</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E83">A575</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c26">p2</font>,<font color="Maroon" title="c25">p1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E81">A573</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E82">A574</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<i><font color="Green" title="E84">A576</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c28">q2</font>,<font color="Maroon" title="c27">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E81">A573</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E82">A574</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E85">A577</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E80">A572</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_1_15"/><b>then </b><i><font color="Green" title="E86">A578</font></i>: 
 ex <font color="Olive" title="b1">tn</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E83">A575</font></i></a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_1_15"/><i><font color="Green" title="E87">A579</font></i>: 
 ex <font color="Olive" title="b1">tm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E84">A576</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E85">A577</font></i></a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_2_1_1_1_15"/>
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E78">A570</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E79">A571</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:2_2_1_2_7_1_2_1_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E86">A578</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_15"><i><font color="Green" title="E87">A579</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E78">A580</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E34:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E79">A581</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_16"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_16"/><i><font color="Green" title="E80">A582</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font> &amp; <font color="Maroon" title="c25">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_16"/><i><font color="Green" title="E81">A583</font></i>: 
( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E80">A582</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_16"/>
( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E81">A583</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_1_16"/><b>then </b><i><font color="Green" title="E82">A584</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_1_16"/><i><font color="Green" title="E83">A585</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E71">A495</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E76">A512</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E82">A584</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E10:2_2_1_2_7_1_2_1_1_1_16"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E81">A583</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E71">A495</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E80">A582</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E81">A583</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_16"><i><font color="Green" title="E83">A585</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E80">A586</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_17"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_17"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_17"/><i><font color="Green" title="E81">A587</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">X</font> &amp; <font color="Maroon" title="c24">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_17"/><i><font color="Green" title="E82">A588</font></i>: 
( <font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E81">A587</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_17"/><i><font color="Green" title="E83">A589</font></i>: 
( <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E82">A588</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_1_17"/><i><font color="Green" title="E84">A590</font></i>: 
( <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E83">A589</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q2</font>,<font color="Maroon" title="c26">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E82">A588</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E84">A590</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<a NAME="E10:2_2_1_2_7_1_2_1_1_1_17"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E81">A587</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E82">A588</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_2_1_1_1_17"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_17"><i><font color="Green" title="E83">A589</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E81">A591</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_18"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_18_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_18_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_18_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_18_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_18_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E82">A592</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E79">A581</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_18_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_18_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_18_1_2_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_18_1_2"><i><font color="Green" title="E82">A592</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_18_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E35:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E80">A586</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_1_18_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E37:2_2_1_2_7_1_2_1_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E82">A593</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E83">A594</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E84">A595</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E83">A594</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">l1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c26">l2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_19"/>
<font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_1_19"/><b>then </b><i><font color="Green" title="E85">A596</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_1_19"/><b>then </b><i><font color="Green" title="E86">A597</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">ql</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E87">A598</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_1_19"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E88">A599</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E89">A600</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E85">A596</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E90">A601</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">qq</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E85">A596</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E91">A602</font></i>: 
( <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E86">A597</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E92">A603</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">qq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E87">A598</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E88">A599</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E89">A600</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E90">A601</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E69">A493</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E91">A602</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<a NAME="E15:2_2_1_2_7_1_2_1_1_1_19"/><i><font color="Green" title="E93">A604</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1"/>
( <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/><i><font color="Green" title="E94">A605</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/><b>then </b><i><font color="Green" title="E95">A606</font></i>: 
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E93">A604</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_1"><i><font color="Green" title="E94">A605</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19_3_1_1"><i><font color="Green" title="E95">A606</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_1_19_3_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2"/><b>then </b><i><font color="Green" title="E94">A607</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c24">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_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="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1"/>
( <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><i><font color="Green" title="E95">A608</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"/><i><font color="Green" title="E96">A609</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_1"><i><font color="Green" title="E96">A609</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A610</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A611</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A612</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">A613</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"/><i><font color="Green" title="E100">A614</font></i>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E92">A603</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E91">A602</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A614</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A614</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A615</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E96">A610</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E99">A613</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c30">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E101">A616</font></i>: 
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E100">A615</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E102">A617</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E95">A608</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E97">A611</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"><i><font color="Green" title="E98">A612</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"><i><font color="Green" title="E101">A616</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1_3"><i><font color="Green" title="E102">A617</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A618</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2"><i><font color="Green" title="E94">A607</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E96">A619</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E95">A618</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/>
( <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E95">A618</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E97">A620</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A619</font></i></a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E98">A621</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_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="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A622</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A622</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A622</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E81">A591</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E98">A621</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E99">A623</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E100">A624</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E101">A625</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_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="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E102">A626</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A627</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A628</font></i>: 
 ex <font color="Olive" title="b1">qx</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E92">A603</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E102">A626</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E105">A629</font></i>: 
( <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E97">A620</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c34">q1l</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E106">A630</font></i>: 
( <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A629</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A624</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E107">A631</font></i>: 
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c33">q2</font>,<font color="Maroon" title="c32">q1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E97">A620</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E106">A630</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A632</font></i>: 
 ex <font color="Olive" title="b1">qy</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A619</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E103">A627</font></i></a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E75">A508</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E86">A597</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A629</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E107">A631</font></i></a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E104">A628</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E108">A632</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E99">A623</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E82">A593</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_19"><i><font color="Green" title="E84">A595</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A624</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E101">A625</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E78">A580</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"><i><font color="Green" title="E96">A619</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_1_19_3_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_19_3_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:2_2_1_2_7_1_2_1_1_1_19"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E38:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E39:2_2_1_2_7_1_2_1_1_1"/><i><font color="Green" title="E82">A633</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>set </b><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<b>set </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_20"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_1_20"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_1_20"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_21"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_2_1_2_7_1_2_1_1_1_21"/>
<font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_21"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_21"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_21"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E41:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E20:2_2_1_2"><i><font color="Green" title="E28">A32</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E82">A633</font></i></a>, <a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></i>;<br/><b>thus </b><a NAME="E42:2_2_1_2_7_1_2_1_1_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c24">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:2_2_1_2_7_1_2_1_1_1_22"/><i><font color="Green" title="E83">A634</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green" title="E84">A635</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_1"/><i><font color="Green" title="E85">A636</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>let </b><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_1"/><i><font color="Green" title="E86">A637</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/><i><font color="Green" title="E87">A638</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E88">A639</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E86">A637</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_1_1_1"><i><font color="Green" title="E87">A638</font></i></a></i>;<br/><b>thus </b><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E68">A492</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E86">A637</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_1_1_1"><i><font color="Green" title="E87">A638</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_1_1_1"><i><font color="Green" title="E88">A639</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_1_22_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E35:2_2_1_2"><i><font color="Green" title="E40">A52</font></i></a>, <a class="txt" href="goboard3.html#E53:2_2_1_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E67">A491</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E86">A637</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_1_1_1"><i><font color="Green" title="E87">A638</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_1_1_1"><i><font color="Green" title="E88">A639</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/><b>then </b><i><font color="Green" title="E87">A640</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E85">A636</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/><i><font color="Green" title="E88">A641</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22_1_1_2"><i><font color="Green" title="E87">A640</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/><b>then </b><i><font color="Green" title="E89">A642</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font> &amp; <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E83">A634</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E85">A636</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/><b>then </b><i><font color="Green" title="E90">A643</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_1_1_2"><i><font color="Green" title="E88">A641</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/>
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_1_1_2"><i><font color="Green" title="E89">A642</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_1_22_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E74">A503</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_1"><i><font color="Green" title="E86">A637</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_1_1_2"><i><font color="Green" title="E90">A643</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_2"/><i><font color="Green" title="E85">A644</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font> &amp; <font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2"/><i><font color="Green" title="E86">A645</font></i>: 
( <font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E85">A644</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><i><font color="Green" title="E87">A646</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_2_1_1"><i><font color="Green" title="E87">A646</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E85">A644</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="topreal3.html#T27" title="TOPREAL3:th.27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><i><font color="Green" title="E87">A647</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><b>then </b><i><font color="Green" title="E88">A648</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><i><font color="Green" title="E89">A649</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E87">A647</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><i><font color="Green" title="E90">A650</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E84">A635</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E88">A648</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E89">A649</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><b>then </b><i><font color="Green" title="E91">A651</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/><i><font color="Green" title="E92">A652</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_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="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A651</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A652</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><i><font color="Green" title="E93">A653</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E85">A644</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A650</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><i><font color="Green" title="E93">A653</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A650</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><i><font color="Green" title="E93">A653</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A651</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A652</font></i></a>, <a class="ref" href="topreal1.html#T7" title="TOPREAL1:th.7">TOPREAL1:7</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E85">A644</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A650</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E91">A651</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E92">A652</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><i><font color="Green" title="E93">A654</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E85">A644</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2"><i><font color="Green" title="E86">A645</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A650</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><i><font color="Green" title="E93">A654</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_2_1_2"><i><font color="Green" title="E90">A650</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><i><font color="Green" title="E93">A654</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:2_2_1_2_7_1_2_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E5:2_2_1_2_7_1_2_1_1_1_22"/><b>then </b><i><font color="Green" title="E85">A655</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="goboard2.html#T8" title="GOBOARD2:th.8">GOBOARD2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E85">A655</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T11" title="GOBOARD2:th.11">GOBOARD2:11</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_2"/><i><font color="Green" title="E86">A656</font></i>: 
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E70">A494</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) );<br/><div><i><font color="Green" title="E87">A657</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E83">A634</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"/><i><font color="Green" title="E88">A658</font></i>: 
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"><i><font color="Green" title="E88">A658</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E86">A656</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:2_2_1_2_7_1_2_1_1_1_22_3_2"/><i><font color="Green" title="E88">A659</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><b>consider </b><font color="Maroon" title="c27">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_3_2"/><i><font color="Green" title="E89">A660</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E88">A659</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E87">A657</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E90">A661</font></i>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E91">A662</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> &amp; <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E92">A663</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E93">A664</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E83">A634</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A661</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E92">A663</font></i></a></i>
;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E94">A665</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E86">A656</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E95">A666</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E83">A634</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A661</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E91">A662</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E96">A667</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E97">A668</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E96">A667</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><i><font color="Green" title="E98">A669</font></i>: 
( <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E84">A635</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E95">A666</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E96">A667</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b><i><font color="Green" title="E99">A670</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c30">q</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A661</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E93">A664</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E97">A668</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E69">A493</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E98">A669</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E99">A670</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E86">A656</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E90">A661</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E94">A665</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E97">A668</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><i><font color="Green" title="E98">A669</font></i></a></i>;<br/><b>hence </b><a NAME="E19:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E86">A656</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E90">A671</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp; <font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E91">A672</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E92">A673</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a></i>;<br/><div><i><font color="Green" title="E93">A674</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/>
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A671</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A672</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E94">A675</font></i>: 
( <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_1_22"><i><font color="Green" title="E84">A635</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E89">A660</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A671</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A672</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E92">A673</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lma</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><i><font color="Green" title="E95">A676</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c30">qa1</font>,<font color="Maroon" title="c29">qa</font>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A671</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A672</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E94">A675</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E95">A676</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2"><i><font color="Green" title="E86">A656</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E90">A671</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E91">A672</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E92">A673</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><i><font color="Green" title="E93">A674</font></i></a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_1_22_3_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E43:2_2_1_2_7_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E44:2_2_1_2_7_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E22:2_2_1_2"><i><font color="Green" title="E30">A34</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E46:2_2_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E60">A477</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E47:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E83">A677</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E48:2_2_1_2_7_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E84">A678</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>thus </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E84">A678</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">m1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E63">A481</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E64">A482</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E84">A678</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a></i>
;<br/><b>thus </b><a NAME="E51:2_2_1_2_7_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E83">A677</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_2_1_1_1"><i><font color="Green" title="E84">A678</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_2"/><i><font color="Green" title="E60">A679</font></i>: 
<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E30:2_2_1_2"><i><font color="Green" title="E36">A48</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_2"><i><font color="Green" title="E60">A679</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3"><b>case </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E60">A680</font></i>: 
<font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j1</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c9">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c8">i2</font>;<br/><b>consider </b><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E61">A681</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E5:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E62">A682</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>;<br/><div><i><font color="Green" title="E63">A683</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_1"/>
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E64">A684</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_1"><i><font color="Green" title="E64">A684</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_1"/>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><a NAME="E7:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E64">A685</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_2"/><i><font color="Green" title="E65">A686</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><b>take </b><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font>;<br/><b>take </b><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c8">i2</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_2"><i><font color="Green" title="E65">A686</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E65">A687</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E66">A688</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_3"/>
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_3"/><b>then </b><i><font color="Green" title="E67">A689</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c23">l1</font>, <font color="Maroon" title="c24">l2</font>, <font color="Maroon" title="c25">l3</font>, <font color="Maroon" title="c26">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font> )
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_3"/><b>then </b>
( <font color="Maroon" title="c23">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c24">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c25">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c26">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_3"><i><font color="Green" title="E67">A689</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">l1</font>, <font color="Maroon" title="c23">l2</font>, <font color="Maroon" title="c24">l3</font>, <font color="Maroon" title="c25">l4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_4"/><i><font color="Green" title="E67">A690</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">l1</font>,<font color="Maroon" title="c23">l2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c24">l3</font>,<font color="Maroon" title="c25">l4</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_4"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_4"/><b>then </b>
( <font color="Maroon" title="c22">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c23">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c24">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c25">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_4"><i><font color="Green" title="E67">A690</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">l3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l4</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E12:2_2_1_2_7_1_2_1_1_3"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2"><i><font color="Green" title="E41">A53</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E66">A688</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E65">A687</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>set </b><font color="Maroon" title="c22">lk</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E14:2_2_1_2_7_1_2_1_1_3"/>
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E67">A691</font></i>: 
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E68">A692</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E69">A693</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E41:2_2_1_2"><i><font color="Green" title="E45">A57</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/><i><font color="Green" title="E70">A694</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E69">A693</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E71">A695</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_6"/><i><font color="Green" title="E72">A696</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_6"/><b>then </b><i><font color="Green" title="E73">A697</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a></i>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_6"/>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_6"/><b>then </b>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_6"><i><font color="Green" title="E73">A697</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_6"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_6"><i><font color="Green" title="E72">A696</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E21:2_2_1_2_7_1_2_1_1_3"/><b>then </b>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E22:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E72">A698</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/><a NAME="E23:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E73">A699</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_7"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_7"/><i><font color="Green" title="E74">A700</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_7"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E62">A682</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_7"><i><font color="Green" title="E74">A700</font></i></a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_3_7"/><b>then </b>
( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_7"><i><font color="Green" title="E74">A700</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_7"/>
contradiction
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E74">A701</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_8"/><i><font color="Green" title="E75">A702</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_8"/><b>then </b><i><font color="Green" title="E76">A703</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><b>set </b><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_8"/><b>then </b><i><font color="Green" title="E77">A704</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_8"/><b>then </b><i><font color="Green" title="E78">A705</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E41:2_2_1_2"><i><font color="Green" title="E45">A57</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="goboard3.html#E49:2_2_1_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="goboard3.html#E50:2_2_1_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E52:2_2_1_2"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_8"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E68">A692</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E75">A702</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><b>thus </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E75">A702</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E77">A704</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E68">A692</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E75">A702</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E76">A703</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_8"><i><font color="Green" title="E78">A705</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E75">A706</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_9"/><i><font color="Green" title="E76">A707</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_9"/><b>then </b><i><font color="Green" title="E77">A708</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E63">A683</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E64">A685</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><b>set </b><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font>;<br/><b>set </b><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font>;<br/><b>set </b><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_9"/>
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E77">A708</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_9"/><b>then </b><i><font color="Green" title="E78">A709</font></i>: 
( <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c12">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E76">A707</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E77">A708</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_9"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E77">A708</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_9"/>
<font color="Maroon" title="c25">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E76">A707</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E77">A708</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_9"><i><font color="Green" title="E78">A709</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E26:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E76">A710</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_10"/><i><font color="Green" title="E77">A711</font></i>: 
( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_10"/><i><font color="Green" title="E78">A712</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_10"><i><font color="Green" title="E77">A711</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_3_10"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_10"><i><font color="Green" title="E77">A711</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_10"><i><font color="Green" title="E78">A712</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_10"><i><font color="Green" title="E77">A711</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_10"><i><font color="Green" title="E78">A712</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:2_2_1_2_7_1_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_11"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_11"/>
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_3_11"/><b>then </b><i><font color="Green" title="E77">A713</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_11_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_11_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_11_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E73">A699</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_11"><i><font color="Green" title="E77">A713</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_11_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_11_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_11_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2"><i><font color="Green" title="E34">A44</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_11"><i><font color="Green" title="E77">A713</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E73">A699</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_11"><i><font color="Green" title="E77">A713</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI: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="E5:2_2_1_2_7_1_2_1_1_3_11"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E28:2_2_1_2_7_1_2_1_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E29:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E22:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E72">A698</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E30:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E77">A714</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E78">A715</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b><i><font color="Green" title="E79">A716</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b><i><font color="Green" title="E80">A717</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E81">A718</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E82">A719</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E80">A717</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E83">A720</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E80">A717</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E84">A721</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E85">A722</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_3_12"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E78">A715</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b><i><font color="Green" title="E86">A723</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c26">u2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E78">A715</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E81">A718</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E82">A719</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E83">A720</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E84">A721</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E85">A722</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E16:2_2_1_2_7_1_2_1_1_3_12"/>
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b><i><font color="Green" title="E87">A724</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E79">A716</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E81">A718</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E82">A719</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E83">A720</font></i></a></i>;<br/>
<a NAME="E18:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b><i><font color="Green" title="E88">A725</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:2_2_1_2_7_1_2_1_1_3_12"/><i><font color="Green" title="E89">A726</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">u</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E82">A719</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E87">A724</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E20:2_2_1_2_7_1_2_1_1_3_12"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">u</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E81">A718</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E83">A720</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E88">A725</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/>
<b>hence </b><a NAME="E21:2_2_1_2_7_1_2_1_1_3_12"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E82">A719</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E86">A723</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3_12"><i><font color="Green" title="E89">A726</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E31:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13"/><i><font color="Green" title="E78">A727</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_3_13"/><b>then </b><i><font color="Green" title="E79">A728</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_3_13"/><i><font color="Green" title="E80">A729</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_13"/><i><font color="Green" title="E81">A730</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_3_13"/><b>then </b><i><font color="Green" title="E82">A731</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_3_13"/><i><font color="Green" title="E83">A732</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_3_13"/><i><font color="Green" title="E84">A733</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_3_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_3_13"/><b>then </b><i><font color="Green" title="E85">A734</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E84">A733</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_1"/><i><font color="Green" title="E86">A735</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="ref" href="goboard2.html#T4" title="GOBOARD2:th.4">GOBOARD2:4</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_13_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_1"><i><font color="Green" title="E86">A735</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2"/><b>then </b><i><font color="Green" title="E86">A736</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_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="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><i><font color="Green" title="E87">A737</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"/><i><font color="Green" title="E88">A738</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A737</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1_1"><i><font color="Green" title="E88">A738</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E88">A739</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E89">A740</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A741</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E81">A730</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A737</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A742</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><i><font color="Green" title="E92">A743</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A737</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E90">A741</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E83">A732</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E84">A733</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A737</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A744</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E79">A728</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E87">A737</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E84">A733</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A745</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><i><font color="Green" title="E95">A746</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font> &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c24">u1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E71">A695</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E80">A729</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E83">A732</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E92">A743</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E93">A744</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E94">A745</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E95">A746</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A747</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E88">A739</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E89">A740</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E91">A742</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E92">A743</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E20:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E80">A729</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E82">A731</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E83">A732</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E22:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E23:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_1"><i><font color="Green" title="E96">A747</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13_1_2"><i><font color="Green" title="E86">A736</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E87">A748</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_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="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E88">A749</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E89">A750</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A749</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E90">A751</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E84">A733</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A749</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A752</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A749</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E92">A753</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A749</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A751</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A751</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E93">A754</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A751</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A755</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E95">A756</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font> &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E96">A757</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font> &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A755</font></i></a></i>;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E97">A758</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A754</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A756</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A757</font></i></a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A759</font></i>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A756</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c27">lg</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E19:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A760</font></i>: 
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A756</font></i></a>, <a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A758</font></i></a></i>;<br/><a NAME="E20:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E100">A761</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c25">u1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E78">A727</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E80">A729</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E82">A731</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E83">A732</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A750</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A752</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E92">A753</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A756</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A757</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E21:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c26">u2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E69">A693</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A757</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A759</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A760</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>hence </b><a NAME="E22:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A752</font></i></a>, <a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A756</font></i></a>, <a class="txt" href="goboard3.html#E20:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_1"><i><font color="Green" title="E100">A761</font></i></a>, <a class="ref" href="topreal1.html#T14" title="TOPREAL1:th.14">TOPREAL1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A762</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"><i><font color="Green" title="E87">A748</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E89">A763</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E82">A731</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A762</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E90">A764</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E81">A730</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A762</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E91">A765</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E88">A762</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E92">A766</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E93">A767</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E82">A731</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E84">A733</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E92">A766</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E79">A728</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E90">A764</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E91">A765</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E92">A766</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E93">A767</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E11:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E30:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E77">A714</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_13"><i><font color="Green" title="E85">A734</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2_1_1_2"><i><font color="Green" title="E89">A763</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_13_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_13_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E32:2_2_1_2_7_1_2_1_1_3"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E78">A768</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E79">A769</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <b>and </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E80">A770</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E4:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E81">A771</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E79">A769</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c29">lp</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c30">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_3_14"/>
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_3_14"/><b>then </b><i><font color="Green" title="E82">A772</font></i>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E79">A769</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E83">A773</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c25">p1</font>,<font color="Maroon" title="c26">p2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E81">A771</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E82">A772</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<i><font color="Green" title="E84">A774</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E81">A771</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E82">A772</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E85">A775</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E80">A770</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_3_14"/><b>then </b><i><font color="Green" title="E86">A776</font></i>: 
 ex <font color="Olive" title="b1">tn</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E83">A773</font></i></a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_3_14"/><i><font color="Green" title="E87">A777</font></i>: 
 ex <font color="Olive" title="b1">tm</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E84">A774</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E85">A775</font></i></a></i>;<br/>
<a NAME="E14:2_2_1_2_7_1_2_1_1_3_14"/>
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E78">A768</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E79">A769</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:2_2_1_2_7_1_2_1_1_3_14"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E86">A776</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_14"><i><font color="Green" title="E87">A777</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E78">A778</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E34:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E79">A779</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_15"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_15"/><i><font color="Green" title="E80">A780</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font> &amp; <font color="Maroon" title="c25">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_15"/><i><font color="Green" title="E81">A781</font></i>: 
( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E80">A780</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_3_15"/>
( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E81">A781</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E7:2_2_1_2_7_1_2_1_1_3_15"/><b>then </b><i><font color="Green" title="E82">A782</font></i>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_3_15"/><i><font color="Green" title="E83">A783</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E71">A695</font></i></a>, <a class="txt" href="goboard3.html#E26:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E76">A710</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E82">A782</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E10:2_2_1_2_7_1_2_1_1_3_15"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E81">A781</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E71">A695</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E80">A780</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E81">A781</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_15"><i><font color="Green" title="E83">A783</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E80">A784</font></i>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_16"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">ls</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_16"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_16"/><i><font color="Green" title="E81">A785</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">X</font> &amp; <font color="Maroon" title="c24">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_16"/><i><font color="Green" title="E82">A786</font></i>: 
( <font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E81">A785</font></i></a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_3_16"/><i><font color="Green" title="E83">A787</font></i>: 
( <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E82">A786</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_3_16"/><i><font color="Green" title="E84">A788</font></i>: 
( <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E83">A787</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c27">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E82">A786</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E84">A788</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<a NAME="E10:2_2_1_2_7_1_2_1_1_3_16"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E81">A785</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E82">A786</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:2_2_1_2_7_1_2_1_1_3_16"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_16"><i><font color="Green" title="E83">A787</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E81">A789</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_17"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_17_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_17_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_17_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_17_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_17_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_17_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_17_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_17_1_2"/><b>then </b><i><font color="Green" title="E82">A790</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E79">A779</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_17_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_17_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_17_1_2_1"/><b>then </b>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_17_1_2"><i><font color="Green" title="E82">A790</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_17_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E35:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E80">A784</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_17_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E37:2_2_1_2_7_1_2_1_1_3"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E82">A791</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E83">A792</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E84">A793</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E83">A792</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">l1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c26">l2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> ;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_18"/>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:2_2_1_2_7_1_2_1_1_3_18"/><b>then </b><i><font color="Green" title="E85">A794</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:2_2_1_2_7_1_2_1_1_3_18"/><b>then </b><i><font color="Green" title="E86">A795</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c28">ql</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/>
<a NAME="E8:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E87">A796</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:2_2_1_2_7_1_2_1_1_3_18"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E88">A797</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<a NAME="E11:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E89">A798</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E85">A794</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E90">A799</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">qq</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E85">A794</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/>
<a NAME="E13:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E91">A800</font></i>: 
( <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">qq</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E86">A795</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<i><font color="Green" title="E92">A801</font></i>:  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>,<font color="Maroon" title="c27">qq</font>
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E87">A796</font></i></a>, <a class="txt" href="goboard3.html#E10:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E88">A797</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E89">A798</font></i></a>, <a class="txt" href="goboard3.html#E12:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E90">A799</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E69">A693</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E91">A800</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/>
<a NAME="E15:2_2_1_2_7_1_2_1_1_3_18"/><i><font color="Green" title="E93">A802</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1"/>
( <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/><i><font color="Green" title="E94">A803</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/><b>then </b><i><font color="Green" title="E95">A804</font></i>: 
( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E93">A802</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_1"><i><font color="Green" title="E94">A803</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18_3_1_1"><i><font color="Green" title="E95">A804</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_18_3_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2"/><b>then </b><i><font color="Green" title="E94">A805</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c24">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_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="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1"/>
( <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><i><font color="Green" title="E95">A806</font></i>: 
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"/><i><font color="Green" title="E96">A807</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2"><i><font color="Green" title="E33">A37</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_1"><i><font color="Green" title="E96">A807</font></i></a>, <a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E96">A808</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="goboard2.html#T9" title="GOBOARD2:th.9">GOBOARD2:9</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A809</font></i>: 
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A810</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">A811</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"/><i><font color="Green" title="E100">A812</font></i>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E92">A801</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E91">A800</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A812</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_2"><i><font color="Green" title="E100">A812</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A813</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E96">A808</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E99">A811</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c30">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E101">A814</font></i>: 
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E100">A813</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E102">A815</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/><b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2"><i><font color="Green" title="E25">A29</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="goboard3.html#E19:2_2_1_2"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E95">A806</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E97">A809</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"><i><font color="Green" title="E98">A810</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"><i><font color="Green" title="E101">A814</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1_3"><i><font color="Green" title="E102">A815</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E95">A816</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2"><i><font color="Green" title="E94">A805</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/>
<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E96">A817</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E95">A816</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/>
( <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E95">A816</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E97">A818</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E96">A817</font></i></a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E98">A819</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_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="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A820</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A820</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A820</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E36:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E81">A789</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E98">A819</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E9:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E99">A821</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E100">A822</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E101">A823</font></i>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_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="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E102">A824</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c31">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span>;<br/><b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A825</font></i>: 
( <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A826</font></i>: 
 ex <font color="Olive" title="b1">qx</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E14:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E92">A801</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E102">A824</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E105">A827</font></i>: 
( <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E97">A818</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c34">q1l</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green" title="E106">A828</font></i>: 
( <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A827</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A822</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E107">A829</font></i>: 
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c32">q1</font>,<font color="Maroon" title="c33">q2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E97">A818</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E106">A828</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A830</font></i>: 
 ex <font color="Olive" title="b1">qy</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E96">A817</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E103">A825</font></i></a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c27">qq</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E25:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E75">A706</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E86">A795</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E105">A827</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E107">A829</font></i></a></i>;<br/><b>hence </b><a NAME="E13:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E104">A826</font></i></a>, <a class="txt" href="goboard3.html#E11:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1_1"><i><font color="Green" title="E108">A830</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E99">A821</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font> &amp; <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E82">A791</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_18"><i><font color="Green" title="E84">A793</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E100">A822</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E101">A823</font></i></a>, <a class="ref" href="goboard2.html#T10" title="GOBOARD2:th.10">GOBOARD2:10</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E33:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E78">A778</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"><i><font color="Green" title="E96">A817</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_3_18_3_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_18_3_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:2_2_1_2_7_1_2_1_1_3_18"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c24">m</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E38:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 <b>by </b><i><a class="ref" href="goboard2.html#T6" title="GOBOARD2:th.6">GOBOARD2:6</a></i>;<br/><a NAME="E39:2_2_1_2_7_1_2_1_1_3"/><i><font color="Green" title="E82">A831</font></i>: 
<font color="Maroon" title="c20">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>set </b><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<b>set </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_19"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ;<br/>

<a NAME="E2:2_2_1_2_7_1_2_1_1_3_19"/><b>then </b>
( <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E3:2_2_1_2_7_1_2_1_1_3_19"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_19"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_20"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_2_1_2_7_1_2_1_1_3_20"/>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_20"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_20"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_20"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E41:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="goboard3.html#E20:2_2_1_2"><i><font color="Green" title="E28">A32</font></i></a>, <a class="txt" href="goboard3.html#E39:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E82">A831</font></i></a>, <a class="ref" href="goboard2.html#T13" title="GOBOARD2:th.13">GOBOARD2:13</a></i>;<br/><b>thus </b><a NAME="E42:2_2_1_2_7_1_2_1_1_3"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c23">lg</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c24">lf</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span> ;<br/>
<a NAME="E1:2_2_1_2_7_1_2_1_1_3_21"/><i><font color="Green" title="E83">A832</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green" title="E84">A833</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_1"/><i><font color="Green" title="E85">A834</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>let </b><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_1"/><i><font color="Green" title="E86">A835</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/><i><font color="Green" title="E87">A836</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E88">A837</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E86">A835</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><i><font color="Green" title="E87">A836</font></i></a></i>;<br/><b>thus </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E40:2_2_1_2"><i><font color="Green" title="E44">A56</font></i></a>, <a class="txt" href="goboard3.html#E51:2_2_1_2"><i><font color="Green" title="E55">A67</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E68">A692</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E86">A835</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><i><font color="Green" title="E87">A836</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><i><font color="Green" title="E88">A837</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><b>thus </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E34:2_2_1_2"><i><font color="Green" title="E39">A51</font></i></a>, <a class="txt" href="goboard3.html#E54:2_2_1_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E67">A691</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E86">A835</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><i><font color="Green" title="E87">A836</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_1_1_1_1"><i><font color="Green" title="E88">A837</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/><b>then </b><i><font color="Green" title="E87">A838</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E85">A834</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/><i><font color="Green" title="E88">A839</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 ;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"><i><font color="Green" title="E87">A838</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/><b>then </b><i><font color="Green" title="E89">A840</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font> &amp; <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E83">A832</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E85">A834</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/><b>then </b><i><font color="Green" title="E90">A841</font></i>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"><i><font color="Green" title="E88">A839</font></i></a>, <a class="ref" href="goboard2.html#T5" title="GOBOARD2:th.5">GOBOARD2:5</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"><i><font color="Green" title="E89">A840</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E24:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E74">A701</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_1"><i><font color="Green" title="E86">A835</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_1_1_1_2"><i><font color="Green" title="E90">A841</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_1"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">c1</font> )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_2"/><i><font color="Green" title="E85">A842</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font> &amp; <font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2"/><i><font color="Green" title="E86">A843</font></i>: 
( <font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E85">A842</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><i><font color="Green" title="E87">A844</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="ref" href="topreal3.html#T25" title="TOPREAL3:th.25">TOPREAL3:25</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"><i><font color="Green" title="E87">A844</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E85">A842</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="topreal3.html#T27" title="TOPREAL3:th.27">TOPREAL3:27</a></i>;<br/><b>hence </b><a NAME="E8:2_2_1_2_7_1_2_1_1_3_21_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><i><font color="Green" title="E87">A845</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A846</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><i><font color="Green" title="E89">A847</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E87">A845</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><i><font color="Green" title="E90">A848</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E84">A833</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E88">A846</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E89">A847</font></i></a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><b>then </b><i><font color="Green" title="E91">A849</font></i>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/><i><font color="Green" title="E92">A850</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_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="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E91">A849</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E92">A850</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><i><font color="Green" title="E93">A851</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E85">A842</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E90">A848</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A851</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E90">A848</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A851</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E91">A849</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E92">A850</font></i></a>, <a class="ref" href="topreal1.html#T7" title="TOPREAL1:th.7">TOPREAL1:7</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E85">A842</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E90">A848</font></i></a></i>;<br/><b>hence </b><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E91">A849</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E92">A850</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><i><font color="Green" title="E93">A852</font></i>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E85">A842</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2"><i><font color="Green" title="E86">A843</font></i></a></i>;<br/><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E90">A848</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><i><font color="Green" title="E93">A852</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"><i><font color="Green" title="E90">A848</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><i><font color="Green" title="E93">A852</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:2_2_1_2_7_1_2_1_1_3_21_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E5:2_2_1_2_7_1_2_1_1_3_21"/><b>then </b><i><font color="Green" title="E85">A853</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="goboard2.html#T8" title="GOBOARD2:th.8">GOBOARD2:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font> )
 </span><b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E85">A853</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_1"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E21:2_2_1_2"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="goboard2.html#T11" title="GOBOARD2:th.11">GOBOARD2:11</a></i>;<br/><b>hence </b><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"/><i><font color="Green" title="E86">A854</font></i>: 
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E18:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E70">A694</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; ( for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) );<br/><div><i><font color="Green" title="E87">A855</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>;<br/><b>thus </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E83">A832</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/>

<b>let </b><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/><i><font color="Green" title="E88">A856</font></i>: 
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ;<br/>

<a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E23:2_2_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"><i><font color="Green" title="E88">A856</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="goboard3.html#E28:2_2_1_2"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A854</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:2_2_1_2_7_1_2_1_1_3_21_3_1_2"/><i><font color="Green" title="E88">A857</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><b>consider </b><font color="Maroon" title="c27">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"/><i><font color="Green" title="E89">A858</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E88">A857</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E87">A855</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_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="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E90">A859</font></i>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E91">A860</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E92">A861</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E93">A862</font></i>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E83">A832</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E90">A859</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E92">A861</font></i></a></i>
;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a></i>;<br/><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E94">A863</font></i>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A854</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E8:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E95">A864</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E83">A832</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E90">A859</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E91">A860</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E96">A865</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E12:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E97">A866</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E96">A865</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E15:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><i><font color="Green" title="E98">A867</font></i>: 
( <font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E84">A833</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E95">A864</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E96">A865</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E16:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E99">A868</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c30">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E90">A859</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E93">A862</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E97">A866</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lq</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E17:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="goboard3.html#E17:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E69">A693</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E98">A867</font></i></a>, <a class="txt" href="goboard3.html#E16:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E99">A868</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>;<br/><a NAME="E18:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c26">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A854</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E90">A859</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E94">A863</font></i></a>, <a class="txt" href="goboard3.html#E13:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E97">A866</font></i></a>, <a class="txt" href="goboard3.html#E15:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><i><font color="Green" title="E98">A867</font></i></a></i>;<br/><b>hence </b><a NAME="E19:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A854</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><div class="add"><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E90">A869</font></i>: 
( <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp; <font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E91">A870</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E92">A871</font></i>: 
<font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a></i>;<br/><div><i><font color="Green" title="E93">A872</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/>
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A869</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A870</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><a NAME="E8:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E94">A873</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2_1_2_7_1_2_1_1_3_21"><i><font color="Green" title="E84">A833</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E89">A858</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A869</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A870</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E92">A871</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c31">lma</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> ;<br/><a NAME="E9:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><i><font color="Green" title="E95">A874</font></i>: 
( <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c29">qa</font>,<font color="Maroon" title="c30">qa1</font>
<b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A869</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A870</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="goboard3.html#E8:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E94">A873</font></i></a>, <a class="txt" href="goboard3.html#E9:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E95">A874</font></i></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></i>
;<br/><a NAME="E11:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2"><i><font color="Green" title="E86">A854</font></i></a>, <a class="txt" href="goboard3.html#E3:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E90">A869</font></i></a>, <a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E91">A870</font></i></a>, <a class="txt" href="goboard3.html#E6:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E92">A871</font></i></a>, <a class="txt" href="goboard3.html#E7:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><i><font color="Green" title="E93">A872</font></i></a></i>;<br/><b>hence </b><a NAME="E12:2_2_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:2_2_1_2_7_1_2_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2_2_1_2_7_1_2_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c21">g</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E43:2_2_1_2_7_1_2_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E3:2_2_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E44:2_2_1_2_7_1_2_1_1_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E22:2_2_1_2"><i><font color="Green" title="E30">A34</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="goboard3.html#E8:2_2_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>
;<br/><br/><a NAME="E46:2_2_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E60">A680</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E47:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E83">A875</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E48:2_2_1_2_7_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E84">A876</font></i>: 
( <font color="Maroon" title="c19">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">l</font></span>)</span>,<font color="Maroon" title="c8">i2</font>
<b>by </b><i><a class="txt" href="goboard3.html#E4:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E61">A681</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E84">A876</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E32:2_2_1_2"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="goboard3.html#E1:2_2_1_2_7_1_2"><i><font color="Green" title="E59">A476</font></i></a></i>
;<br/><br/><b>thus </b><a NAME="E50:2_2_1_2_7_1_2_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <b>by </b><i><a class="txt" href="goboard3.html#E2:2_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard3.html#E5:2_2_1_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="goboard3.html#E24:2_2_1_2"><i><font color="Green" title="E32">A36</font></i></a>, <a class="txt" href="goboard3.html#E47:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E83">A875</font></i></a>, <a class="txt" href="goboard3.html#E48:2_2_1_2_7_1_2_1_1_3"><i><font color="Green" title="E84">A876</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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="E3:2_2_1_2_7_1_2"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E56:2_2_1_2"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="goboard3.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="goboard3.html#E2:2"><i><font color="Green" title="E4">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:2"/>
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> ) ) &amp; <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> implies  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Olive" title="b1">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) )
 ;<br/>


</div>
