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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</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="b2">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="b3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b3">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</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">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">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="c4">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">aA</font> = <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">AB</font> = <font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">aAB</font> = <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">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="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<a NAME="E2:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</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">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/>
<a NAME="E3:1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <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#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<a NAME="E4:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A4</font></span>: 
 <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#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></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="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/>
<a NAME="E5:1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
 <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#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</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">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<a NAME="E6:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A6</font></span>: 
(  <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#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</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="c3">A</font> &amp;  <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#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E8">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)</span><br/><a NAME="E2:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T18" title="MATRIXR1:th.18">MATRIXR1:18</a></span>;<br/><a NAME="E3:1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E4:1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A9</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E7:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E5:1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>) = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">j</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c8">i</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">j</font>)</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c8">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="fvsum_1.html#T68" title="FVSUM_1:th.68">FVSUM_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="matrixr1.html#T20" title="MATRIXR1:th.20">MATRIXR1:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E5:1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E7:1_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:1"/>
(  <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#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E5:1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E7:1"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
