<?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="E1:23"/><i><font color="Green">E52</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 &amp;  <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>set </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> ;<br/>
<a NAME="E2:23"/><i><font color="Green">E53</font></i>: 
(  <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> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</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>  <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> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j</font>,<font color="Olive">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> ) ) &amp; ( for <font color="Olive">i</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">j</font>,<font color="Olive">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> holds <br/><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">j</font>,<font color="Olive">i</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E3:23"/><i><font color="Green">E54</font></i>: 
 <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>, </i>;<br/>
<a NAME="E4:23"/><i><font color="Green">E55</font></i>: 
 <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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:23"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <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> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</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>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j</font>,<font color="Olive">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> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> ) ) &amp; ( for <font color="Olive">i</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">j</font>,<font color="Olive">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> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">j</font>,<font color="Olive">i</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green">E62</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <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>, , </i>;<br/>
<a NAME="E7:23"/>
(  <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> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </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"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E9:23"/>
for <font color="Olive">i</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">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E64</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>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:23_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> <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/>
<a NAME="E3:23_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<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#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</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="E4:23_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<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">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:23"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
