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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">B</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="b3">C</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">B</font> <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="Olive" title="b3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">B</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="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>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">C</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</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="c4">C</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font> <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="c4">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="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">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</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="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>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</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="c4">C</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font> <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="c4">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="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">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">B</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="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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E2:20"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
(  <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="c4">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="c4">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="c4">C</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</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">C</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_4.html#D1" title="MATRIX_4:def.1">MATRIX_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="matrix_4.html#T63" title="MATRIX_4:th.63">MATRIX_4:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2" title="MATRIX_3:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="matrixr2.html#T15" target="_self" title="MATRIXR2:th.15">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix_4.html#K1" title="MATRIX_4:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_4.html#D1" title="MATRIX_4:def.1">MATRIX_4:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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