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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:28_1_1"/><i><font color="Green">E40</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:28_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:28_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Olive">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:28_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:28_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:28_1_1_1"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T113">FINSEQ_3:113</a></i>;<br/>
<b>hence </b><a NAME="E5:28_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">j</font>,<font color="Olive">x1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">j</font>,<font color="Olive">x2</font>] holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
;<br/>
<a NAME="E8:28_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">j</font>,<font color="Olive">x</font>]
 
<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">k</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:28_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:28_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> =  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_3.html#T114">FINSEQ_3:114</a></i>;<br/>
<a NAME="E4:28_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:28_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E7:28_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">j</font>,<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">N</font> being    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>,<font color="Maroon">m</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:28_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">j</font>,<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">k</font>,<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="matrix_1.html#S2">MATRIX_1:sch 2</a>(, );<br/></i>
<a NAME="E11:28_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:28_1_1"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<a NAME="E13:28_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">k</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Olive">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:28_1_1_3"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:28_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_3"/>
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:28_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; not <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">k</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">k</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Olive">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:28_1_1_4"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:28_1_1_4"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:28_1_1_4"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:28_1_1_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E5:28_1_1_4"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1_4"/>
( <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:28_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Olive">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_1_1_5"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:28_1_1_5"/><b>then </b><i><font color="Green">E77</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:28_1_1_5"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:28_1_1_5"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:28_1_1_5"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:28_1_1_5"/><i><font color="Green">E79</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_5_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_5_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_5_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_5_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:28_1_1_5_1_1_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_5_1_1_1"/>
(  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_5_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_5_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_5_1_1_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:28_1_1_5_1_1_2_1"/><i><font color="Green">E86</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E4:28_1_1_5_1_1_2_1"/><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T120">FINSEQ_3:120</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_5_1_1_2"/>
(  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:28_1_1_5"/>
(  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">N</font> as   <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E17:28_1_1"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E18:28_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_6"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:28_1_1_6"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:28_1_1_6"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:28_1_1_6"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">L</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font>, <font color="Maroon">lg</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon">X</font> =  <a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font>;<br/><b>set </b><font color="Maroon">xg</font> =  <a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_6_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_6_1"/><i><font color="Green">E121</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lg</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:28_1_1_6_1"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E4:28_1_1_6_1"/><b>then </b><i><font color="Green">E123</font></i>: 
( ( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> or <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E5:28_1_1_6_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:28_1_1_6_1"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:28_1_1_6_1"/><b>then </b><i><font color="Green">E124</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E8:28_1_1_6_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:28_1_1_6_1"/><b>then </b><i><font color="Green">E125</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:28_1_1_6_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:28_1_1_6_1"/><b>then </b><i><font color="Green">E126</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:28_1_1_6_1"/><b>then </b><i><font color="Green">E127</font></i>: 
( <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:28_1_1_6_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:28_1_1_6_1"/><b>then </b>
( <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:28_1_1_6_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard1.html#T2" target="_self">Th2</a>, , </i>;<br/><b>hence </b><a NAME="E16:28_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E6:28_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard1.html#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:28_1_1_6"/>
 <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_7"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span>
 ;<br/><a NAME="E2:28_1_1_7"/><b>then </b><i><font color="Green">E130</font></i>: 
(  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:28_1_1_7"/><i><font color="Green">E131</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:28_1_1_7"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:28_1_1_7"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:28_1_1_7"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_7"/>
 <a href="goboard1.html#K3" target="_self">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">L</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font>, <font color="Maroon">lg</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon">X</font> =  <a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font>;<br/><b>set </b><font color="Maroon">xg</font> =  <a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font>;<br/><a NAME="E2:28_1_1_8"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:28_1_1_8"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><b>assume </b><a NAME="E4:28_1_1_8"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E5:28_1_1_8"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font> is <a href="goboard1.html#V1" target="_self">increasing</a>
 <b>by </b><i>, , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_8_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:28_1_1_8_1"/><i><font color="Green">E137</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lg</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:28_1_1_8_1"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E4:28_1_1_8_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:28_1_1_8_1"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:28_1_1_8_1"/><b>then </b><i><font color="Green">E139</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E6:28_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E7:28_1_1_8_1"/><i><font color="Green">E140</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:28_1_1_8_1"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:28_1_1_8_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:28_1_1_8_1"/><b>then </b><i><font color="Green">E142</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:28_1_1_8_1"/><b>then </b><i><font color="Green">E143</font></i>: 
( <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E12:28_1_1_8_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:28_1_1_8_1"/><b>then </b>
( <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">lg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lg</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E7:28_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E14:28_1_1_8_1"/><b>then </b><i><font color="Green">E144</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">lg</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E7:28_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>;<br/><a NAME="E15:28_1_1_8_1"/><i><font color="Green">E145</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">lg</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_8_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_8_1_1_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_8_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E148</font></i>: <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">lg</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard1.html#E12:28_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
;<br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">lg</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard1.html#E11:28_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
;<br/><b>hence </b><a NAME="E5:28_1_1_8_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="goboard1.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_8_1_1_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><i><font color="Green">E150</font></i>: <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">lg</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard1.html#T2" target="_self">Th2</a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard1.html#E10:28_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="goboard1.html#E11:28_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T120">FINSEQ_3:120</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_8_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_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="E1:28_1_1_8_1_1_2_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_8_1_1_2_2_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_8_1_1_2_2_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:28_1_1_8_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">lg</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
;<br/><b>hence </b><a NAME="E5:28_1_1_8_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="goboard1.html#E7:28_1_1"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="goboard1.html#E12:28_1_1"><i><font color="Green">E52</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_8_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_8_1_1_2_2_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_8_1_1_2_2_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">lg</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , , <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard1.html#T2" target="_self">Th2</a>, <a class="txt" href="goboard1.html#E8:28_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard1.html#E10:28_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T120">FINSEQ_3:120</a></i>
;<br/><b>hence </b><a NAME="E4:28_1_1_8_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="goboard1.html#E7:28_1_1"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="goboard1.html#E12:28_1_1"><i><font color="Green">E52</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="E4:28_1_1_8_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" target="_self">Y_axis</a> <font color="Maroon">L</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:28_1_1_8"/>
 <a href="goboard1.html#K3" target="_self">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> is <a href="goboard1.html#V1" target="_self">increasing</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_9"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span>
 ;<br/><a NAME="E2:28_1_1_9"/><b>then </b><i><font color="Green">E156</font></i>: 
(  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">k</font> or  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:28_1_1_9"/><i><font color="Green">E157</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:28_1_1_9"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:28_1_1_9"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:28_1_1_9"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_9"/>
 <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> is <a href="goboard1.html#V1" target="_self">increasing</a>
 <b>by </b><i><a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as  <a href="relat_1.html#V3">V6</a> <a href="goboard1.html#V3" target="_self">X_equal-in-line</a> <a href="goboard1.html#V4" target="_self">Y_equal-in-column</a> <a href="goboard1.html#V5" target="_self">Y_increasing-in-line</a> <a href="goboard1.html#V6" target="_self">X_increasing-in-column</a> <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, , , , , , , , </i>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E24:28_1_1"/><i><font color="Green">E158</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>

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

<b>assume </b><a NAME="E25:28_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E26:28_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<a NAME="E28:28_1_1"/><i><font color="Green">E161</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E29:28_1_1"/><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_10"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:28_1_1_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_10"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:28"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard1.html#E1:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard1.html#E6:28_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E31:28_1_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E3:28_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard1.html#E5:28_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
