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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">j</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#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">B</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> )
 ;<br/>

<b>given </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:57"/><i><font color="Green">E30</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>
 ;<br/>

<a NAME="E3:57"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</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="matrix_3.html#K3">+</a> <font color="Maroon">B</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="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font>, <font color="Maroon">b</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/>
<i><font color="Green">E33</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>

<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E7:57"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E1:57"><i><font color="Green">E29</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E8:57"/>
for <font color="Olive">k</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">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> )
 ;<br/>

<a NAME="E2:57_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E1:57"><i><font color="Green">E29</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E3:57_2"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:57_2"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E5:57_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</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">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E6:57"><i><font color="Green">E33</font></i></a>, <a class="txt" href="matrix_4.html#E7:57"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:57_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">k</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">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></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">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="matrix_4.html#E1:57"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:57_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</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="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_4.html#E2:57"><i><font color="Green">E30</font></i></a>, <a class="ref" href="matrix_4.html#T7" target="_self">Th7</a>, <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>
;<br/>
<a NAME="E9:57_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</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"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_4.html#E6:57"><i><font color="Green">E33</font></i></a>, <a class="txt" href="matrix_4.html#E7:57"><i><font color="Green">E34</font></i></a>, <a class="ref" href="matrix_4.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:57_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E12:57_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</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"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="matrix_4.html#E6:57"><i><font color="Green">E33</font></i></a>, <a class="txt" href="matrix_4.html#E7:57"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:57_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>

<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E16:57_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:57_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T21">FVSUM_1:21</a></i>;<br/>
<i><font color="Green">E81</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> = 
 <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">A</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E19:57_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="matrix_4.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_4.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:57_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E21:57_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="matrix_4.html#E1:57_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="matrix_4.html#T9" target="_self">Th9</a>, , , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
 <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K3">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">A</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_4.html#E3:57"><i><font color="Green">E31</font></i></a>, <a class="txt" href="matrix_4.html#E4:57"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
