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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">C</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E19">A1</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span></span><br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></i>;<br/>
<a NAME="E5:22"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E24">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E3:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E4:22"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E25">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">D</font> = <font color="Maroon" title="c3">C</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E7:22"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="matrix_4.html#T62" title="MATRIX_4:th.62">MATRIX_4:62</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T7" title="MATRIX_3:th.7">MATRIX_3:7</a></i>
<br/>.= 
 <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">C</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrixr2.html#T18" target="_self" title="MATRIXR2:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E10:22"/>
<font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E5:22"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="#E6:22"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="matrix_4.html#T8" title="MATRIX_4:th.8">MATRIX_4:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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