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

<b>let </b><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c6">M</font>, <font color="Maroon" title="c7">M1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">K</font>;<br/>













<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  )
 ;<br/>

<a NAME="E2:31"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E4:31"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></i>;<br/>
<a NAME="E5:31"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E3:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:31"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">K</font>
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">M2</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">K</font><b> such that </b><br/><a NAME="E8:31"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c8">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ;<br/>
<a NAME="E9:31"/><i><font color="Green" title="E28">A5</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> holds <br/>( ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">k</font> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">l</font> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> )
 ;<br/>

<a NAME="E2:31_1"/><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M1</font>
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">M1</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">M1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E6:31_1"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix12.html#E3:31_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E7:31_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 ;<br/>

<a NAME="E2:31_1_2"/><i><font color="Green" title="E35">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E1:31_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E8:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="matrix12.html#E6:31_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E4:31_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="matrix12.html#E1:31_1_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E2:31_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font>
 ;<br/>


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

<b>thus </b><a NAME="E8:31_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_3"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:31_1_3"/><i><font color="Green" title="E35">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E1:31_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E8:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="matrix12.html#E6:31_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E4:31_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="matrix12.html#E1:31_1_3"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font>
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E2:31_1_3"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E4:31_1_3"/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font>
 ;<br/>


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

<b>assume </b><a NAME="E9:31_1"/><i><font color="Green" title="E34">A15</font></i>: 
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> )
 ;<br/>

<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E8:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="matrix12.html#E6:31_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="matrix12.html#E4:31_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="matrix12.html#E9:31_1"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E2:31_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E11:31_1"/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:31"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E29">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span>
 ;<br/>

<a NAME="E2:31_2"/><i><font color="Green" title="E30">A17</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E8:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/>
<a NAME="E3:31_2"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="matrix12.html#E2:31_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:31_2"/><b>then </b>
( ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c2">k</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix12.html#E8:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="matrix12.html#E9:31"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="matrix12.html#D4" target="_self" title="MATRIX12:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:31"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="matrix12.html#E6:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/>


</div>
