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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>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E24">A1</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> <font color="Maroon" title="c3">B</font> &amp;  <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="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c2">A</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="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c3">B</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="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="matrix15.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">AB</font> = <font color="Maroon" title="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B'</font>;<br/>
<b>set </b><font color="Maroon" title="c8">BA</font> = <font color="Maroon" title="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">A'</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
(  <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="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or  <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="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/><i><font color="Green" title="E25">A2</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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><a NAME="E2:25_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:25_1_1"/><b>then </b>
( <font color="Maroon" title="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="matrix15.html#E1:25_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>hence </b><a NAME="E4:25_1_1"/>
( <font color="Maroon" title="c2">A</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
 <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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b><i><font color="Green" title="E25">A3</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="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">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="c3">B</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="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">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">B</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="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B'</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">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="c2">A</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><b>hence </b><font color="Maroon" title="c2">A</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c3">B</font> = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c6">B'</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix13.html#T46" title="MATRIX13:th.46">MATRIX13:46</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c3">B</font>
<b>by </b><i><a class="txt" href="matrix15.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="matrix15.html#T19" target="_self" title="MATRIX15:th.19">Th19</a></i>
;<br/><br/><b>thus </b><font color="Maroon" title="c3">B</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">^^</a> <font color="Maroon" title="c2">A</font> = 
 <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B'</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c5">A'</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E2:25_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="matrix13.html#T46" title="MATRIX13:th.46">MATRIX13:46</a></i>
<br/>.= 
<font color="Maroon" title="c3">B</font>
<b>by </b><i><a class="ref" href="matrix15.html#T19" target="_self" title="MATRIX15:th.19">Th19</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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