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

<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font>;<br/>
<a NAME="E1:28_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:28_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</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="c1">v1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</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="c2">v2</font> )
 
<b>by </b><i><a class="ref" href="goboard2.html#D1" target="_self" title="GOBOARD2:def.1">Def1</a></i>;<br/>
<a NAME="E3:28_1_1"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</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="c1">v1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</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="c2">v2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:28_1_1"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <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="c1">v1</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">v2</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/>
<b>thus </b><a NAME="E5:28_1_1"/>
not  <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font> is <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:28_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="goboard2.html#E2:28_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E6:28_1_1"/>
 <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font> is <a href="goboard1.html#V3" title="GOBOARD1:attr.3">X_equal-in-line</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D6" title="GOBOARD1:def.6">GOBOARD1:def 6</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">l</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span>,<font color="Maroon" title="c4">n</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="c6">x</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">l</font>;<br/>
<b>assume </b><a NAME="E2:28_1_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">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"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span>
 ;<br/>

<a NAME="E3:28_1_1_1"/><i><font color="Green" title="E27">A6</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="c5">l</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="c5">l</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> &amp;  <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="c5">l</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="c5">l</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>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:28_1_1_1"/><b>then </b><i><font color="Green" title="E28">A7</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="c5">l</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="c5">l</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">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="E1:28_1_1_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">l</font></span>)</span> &amp; <font color="Maroon" title="c8">j</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">l</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_1_1"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">i</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> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">n</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> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:28_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="goboard2.html#E4:28_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="goboard2.html#E2:28_1_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="goboard2.html#E3:28_1_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c1">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c2">v2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c11">s2</font> = <font color="Maroon" title="c2">v2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:28_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">i</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"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c10">s1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c8">j</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"><font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c11">s2</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:28_1_1_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="goboard2.html#D1" target="_self" title="GOBOARD2:def.1">Def1</a></i>;<br/><a NAME="E5:28_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A10</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">i</font></span>)</span> <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="c9">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c8">j</font></span>)</span> <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="c9">r</font> )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E6:28_1_1_1_1"/>
( <font color="Maroon" title="c5">l</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:28_1_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="goboard2.html#E1:28_1_1_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E7:28_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">l</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:28_1_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="goboard2.html#E1:28_1_1_1_1"><i><font color="Green" title="E29">A8</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="E8:28_1_1_1_1"/><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="c5">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:28_1_1_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="goboard2.html#E5:28_1_1_1_1"><i><font color="Green" title="E31">A10</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="E9:28_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</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="c5">l</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1_1_1"/>
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span>,<font color="Maroon" title="c4">n</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E7:28_1_1"/>
 <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font> is <a href="goboard1.html#V4" title="GOBOARD1:attr.4">Y_equal-in-column</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D7" title="GOBOARD1:def.7">GOBOARD1:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">c</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span>,<font color="Maroon" title="c4">n</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="c6">y</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">c</font>;<br/>
<b>assume </b><a NAME="E2:28_1_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c4">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> <span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:28_1_1_2"/>
(  <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="c5">c</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="c5">c</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> &amp;  <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="c5">c</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="c5">c</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>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:28_1_1_2"/><b>then </b><i><font color="Green" title="E27">A12</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="c5">c</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="c5">c</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">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="E1:28_1_1_2_1"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">c</font></span>)</span> &amp; <font color="Maroon" title="c8">j</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"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">c</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_2_1"/><b>then </b><i><font color="Green" title="E29">A14</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="c4">n</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> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c4">n</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> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:28_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="goboard2.html#E4:28_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="goboard2.html#E2:28_1_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="goboard2.html#E4:28_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c2">v2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c1">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c11">s2</font> = <font color="Maroon" title="c1">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:28_1_1_2_1"/>
( <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">n</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"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c9">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c4">n</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"><font color="Maroon" title="c11">s2</font>,<font color="Maroon" title="c9">r</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E2:28_1_1_2_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="goboard2.html#D1" target="_self" title="GOBOARD2:def.1">Def1</a></i>;<br/><a NAME="E5:28_1_1_2_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c9">r</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c9">r</font> )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E6:28_1_1_2_1"/>
( <font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:28_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="goboard2.html#E1:28_1_1_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E7:28_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E4:28_1_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="goboard2.html#E1:28_1_1_2_1"><i><font color="Green" title="E28">A13</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="E8:28_1_1_2_1"/><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="c5">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font> )
 <b>by </b><i><a class="txt" href="goboard2.html#E1:28_1_1_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="goboard2.html#E5:28_1_1_2_1"><i><font color="Green" title="E30">A15</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="E9:28_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</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="c5">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1_1_2"/>
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <font color="Maroon" title="c1">v1</font>,<font color="Maroon" title="c2">v2</font></span>)</span>,<font color="Maroon" title="c4">n</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/>


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


</div>
