<?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">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">c2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">d1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">d2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/>



<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c1</font>,<font color="Maroon">d1</font> as   <a href="matrix_1.html#NM5">Matrix</a> of 2,<font color="Maroon">K</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c2</font>,<font color="Maroon">d2</font> as   <a href="matrix_1.html#NM5">Matrix</a> of 2,<font color="Maroon">K</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span></span>)</span> as   <a href="matrix_1.html#NM5">Matrix</a> of 2,<font color="Maroon">K</font> ;<br/>
<a NAME="E4:29"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font> as   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font> ;<br/>
<a NAME="E6:29"/><i><font color="Green">E44</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<i><font color="Green">E45</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,1</span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:29_1"/><i><font color="Green">E48</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> 1,2
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<b>hence </b><a NAME="E5:29_1"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:29"/><i><font color="Green">E50</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<i><font color="Green">E51</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,2</span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:29_2"/><i><font color="Green">E52</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> 2,1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">c1</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> 2,2
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<b>hence </b><a NAME="E5:29_2"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:29"/><i><font color="Green">E68</font></i>: 
(  <a href="setwop_2.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> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> &amp;  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">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="E9:29"/><i><font color="Green">E69</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">c2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<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>,1</span>)</span> = 
 <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/>.= 
2
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:29_3"/><i><font color="Green">E76</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> 2,1
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">c2</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<b>hence </b><a NAME="E5:29_3"/>
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">c2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:29"/><i><font color="Green">E78</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">d2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>proof </b></a><div class="add">

<i><font color="Green">E79</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>,2</span>)</span> = 
 <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/>.= 
2
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:29_4"/><i><font color="Green">E80</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> 1,2
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> 2,2
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/>
<b>hence </b><a NAME="E5:29_4"/>
 <a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">d2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:29"/><i><font color="Green">E82</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="matrix_1.html#NM5">Matrix</a> of 2,<font color="Maroon">K</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7">MATRIX_2:7</a></i>;<br/>
<a NAME="E13:29"/>
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:29"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5"><b>now </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="E1:29_5"/><i><font color="Green">E84</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">C</font>
 ;<br/><a NAME="E2:29_5"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2 )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_5_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_5_1_1_1"><b>case </b></a><a NAME="E1:29_5_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> 1,1 = 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">c2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T114">FVSUM_1:114</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_5_1_1_2"><b>case </b></a><a NAME="E1:29_5_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> 1,2 = 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">d2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T114">FVSUM_1:114</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="matrix_1.html#K3">*</a> 1,2
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_5_1_1_3"><b>case </b></a><a NAME="E1:29_5_1_1_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> 2,1 = 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">c2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T114">FVSUM_1:114</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="matrix_1.html#K3">*</a> 2,1
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_5_1_1_4"><b>case </b></a><a NAME="E1:29_5_1_1_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> 2,2 = 
<span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</font></span><a href="rlvect_1.html#K7">*&gt;</a></span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">d2</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T114">FVSUM_1:114</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="matrix_1.html#K3">*</a> 2,2
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_5"/>
<font color="Maroon">C</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">D</font> <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="E16:29"/>
<span class="p1">(<span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c1</font>,<font color="Maroon">d1</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c2</font>,<font color="Maroon">d2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span></span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28">MATRIX_1:28</a></i>;<br/>


</div>
