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<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>, <font color="Maroon">b</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> = <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> as   <a href="matrix_1.html#NM5">Matrix</a> of 1,<font color="Maroon">K</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> as   <a href="matrix_1.html#NM5">Matrix</a> of 1,<font color="Maroon">K</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> as   <a href="matrix_1.html#NM5">Matrix</a> of 1,<font color="Maroon">K</font> ;<br/>
<a NAME="E4:28"/><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> 1 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 &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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 1</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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 1</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> 1 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 &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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 1</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="E6:28"/><i><font color="Green">E44</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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/>.= 
1
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:28_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.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="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><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 = 
<span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T5">MATRIX_2:5</a></i>
;<br/>
<b>hence </b><a NAME="E4:28_1"/>
 <a href="matrix_1.html#K8">Line</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28"/><i><font color="Green">E48</font></i>: 
 <a href="matrix_1.html#K9">Col</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<i><font color="Green">E49</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/>.= 
1
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:28_2"/>
1 <a href="hidden.html#R2">in</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">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
1 <a href="hidden.html#R2">in</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/>
<b>then </b><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 = 
<span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T5">MATRIX_2:5</a></i>
;<br/>
<b>hence </b><a NAME="E5:28_2"/>
 <a href="matrix_1.html#K9">Col</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:28"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1 )
 
<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 1,<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="E10:28"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:28"/><b>then </b><i><font color="Green">E51</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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 1</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="28_3"><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="E1:28_3"/><i><font color="Green">E52</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="E2:28_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 1 )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:28_3"/><b>then </b><i><font color="Green">E59</font></i>: 
( <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 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#T113">FVSUM_1:113</a></i>
<br/>.= 
<span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrix_2.html#T5">MATRIX_2:5</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:28"/>
<span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28">MATRIX_1:28</a></i>;<br/>


</div>
