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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:37"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:37"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/>
<a NAME="E6:37"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">p2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:37"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:37"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E10:37"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E11:37"/><i><font color="Green">E59</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">i2</font>, <font color="Olive">j2</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">i2</font>,<font color="Olive">j2</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">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font></span>)</span> ) )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:37"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <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="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">s3</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:37"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a>, </i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:37"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> holds <br/> ex <font color="Olive">s2</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">n3</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:37"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">s2</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E21:37"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">s4</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E23:37"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">s4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i>, <a class="ref" href="matrixr1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E24:37"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">qq</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E26:37"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">qq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E27:37"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">qq0</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E29:37"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">qq0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qq0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> )
 <b>by </b><i><a class="ref" href="matrixr1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E30:37"/><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , , <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E31:37"/><i><font color="Green">E102</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qq</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E32:37"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font></span>)</span> holds <br/><font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_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"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/>

<a NAME="E2:37_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<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">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E1:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:37_1"/><i><font color="Green">E104</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> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T3" target="_self">Th3</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:37_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">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="matrixr1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:37_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></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">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">p2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:37_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">A</font> <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">p2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, </i>;<br/>
<a NAME="E9:37_1"/>
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , , , , </i>;<br/>
<b>hence </b><a NAME="E10:37_1"/>
<font color="Maroon">qq</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"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E33:37"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrixr1.html#K7" target="_self">*</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="matrixr1.html#E1:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="matrixr1.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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