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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E44">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</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="c2">p</font>
 ;<br/>

<i><font color="Green" title="E45">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">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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"><span class="p3">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#D12" title="MATRIXR1:def.12">MATRIXR1:def 12</a></i>
<br/>.= 
 <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:50_2"/><i><font color="Green" title="E47">A4</font></i>: 
<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"><span class="p3">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span>,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:50_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:50_2"/><i><font color="Green" title="E48">A5</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</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="c1">M</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E1:50"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="matrixr1.html#D10" title="MATRIXR1:def.10">MATRIXR1:def 10</a></i>;<br/><a NAME="E4:50_2"/><b>then </b>
(  <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</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="c1">M</font> )
 <b>by </b><i><a class="ref" href="matrprob.html#T39" target="_self" title="MATRPROB:th.39">Th39</a></i>;<br/><a NAME="E5:50_2"/><b>then </b>
(  <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</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="c1">M</font> )
 <b>by </b><i><a class="ref" href="matrixr1.html#T48" title="MATRIXR1:th.48">MATRIXR1:48</a></i>;<br/><a NAME="E6:50_2"/><b>then </b>
( 1 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrprob.html#E2:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:50_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:50_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c3">i</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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrprob.html#T12" target="_self" title="MATRPROB:th.12">Th12</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c3">i</font> = 
<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"><a href="matrixr1.html#K10" title="MATRIXR1:func.10">LineVec2Mx</a> <font color="Maroon" title="c2">p</font></span>)</span>,1</span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:50_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="matrprob.html#T39" target="_self" title="MATRPROB:th.39">Th39</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T48" title="MATRIXR1:th.48">MATRIXR1:48</a></i>
;<br/><b>hence </b><a NAME="E9:50_2"/>
<font color="Maroon" title="c2">p</font> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E2:50_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="matrixr1.html#D12" title="MATRIXR1:def.12">MATRIXR1:def 12</a></i>;<br/></div>
<b>end;</b></div>


</div>
