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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E52</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</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#D7">MATRIX_1:def 7</a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</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">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:20_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E4:20_1_2"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:20_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:20_1_2_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"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </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> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:20_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:20_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</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">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</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="zfmisc_1.html#T107">ZFMISC_1:107</a></i>;<br/>


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

<b>assume </b><a NAME="E2:20_1_2_1"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:20_1_2_1"/><b>then </b>
( <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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:20_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</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">M</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T107">ZFMISC_1:107</a></i>;<br/>
<a NAME="E6:20_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:20_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:20_1_2"/><b>then </b>
<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="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <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> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T108">ZFMISC_1:108</a></i>;<br/><a NAME="E6:20_1_2"/><b>then </b>
(  <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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T134">ZFMISC_1:134</a></i>;<br/><b>hence </b><a NAME="E7:20_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
