<?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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<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="E1:22"/><i><font color="Green">E31</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">B</font>
 ;<br/>

<b>set </b><font color="Maroon">C</font> = <font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span>;<br/>
<a NAME="E2:22"/><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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</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="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green">E55</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#K6">*</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">B</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green">E56</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#K4">*</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>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, </i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E57</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</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">B</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, </i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E58</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</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="matrix_3.html#K6">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<i><font color="Green">E59</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a>, </i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E8:22"/>
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">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></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"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Olive">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</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#K6">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><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_2"/><i><font color="Green">E63</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"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:22_2"/>
<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">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, , </i>;<br/>
<a NAME="E3:22_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( 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="matrix_1.html#K1">width</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:22_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</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">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:22_2"/><b>then </b><i><font color="Green">E69</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"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a>, </i>;<br/>
<a NAME="E6:22_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></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">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</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><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xo</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">j</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</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/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</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_3.html#D4">MATRIX_3:def 4</a>, , </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<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></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<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></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T92">FVSUM_1:92</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <font color="Maroon">xo</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T83">FVSUM_1:83</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E9:22_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span></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"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</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#K6">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:22"/>
<font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, , , , , </i>;<br/>


</div>
