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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM3">Matrix</a> of <font color="Maroon">n</font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<a NAME="E1:22"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <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#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#T27">MATRIXR1:27</a></i>;<br/>
<a NAME="E2:22"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></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="E3:22"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:22"/>
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"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> holds <br/><span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></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"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <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="22_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="E1:22_1"/><i><font color="Green">E45</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"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span>
 ;<br/>

<a NAME="E2:22_1"/><i><font color="Green">E46</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1"/><i><font color="Green">E47</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></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><a class="ref" href="matrixr1.html#T28">MATRIXR1:28</a></i>;<br/>
<i><font color="Green">E48</font></i>: <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/>
<a NAME="E5:22_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#T27">MATRIXR1:27</a></i>;<br/>
<a NAME="E6:22_1"/><i><font color="Green">E62</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:22_1"/><i><font color="Green">E63</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></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><a class="ref" href="matrixr1.html#T28">MATRIXR1:28</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, , , , </i>
;<br/>
<b>hence </b><a NAME="E9:22_1"/>
<span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></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"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p2"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22"/>
<span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix10.html#K5" target="_self">*</a> <font color="Maroon">M</font></span>)</span></span><a href="matrix10.html#K1" target="_self">:|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="matrixr1.html#K7">*</a> <span class="p1"><a href="matrix10.html#K1" target="_self">|:</a><span class="default"><font color="Maroon">M</font></span><a href="matrix10.html#K1" target="_self">:|</a></span>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
