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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">c</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">AD</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">D</font>;<br/><b>let </b><font color="Maroon" title="c6">pD</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">D</font>;<br/>











<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E18">A1</font></i>: 
not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">AD</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="laplace.html#NK4" title="LAPLACE:NK.4">RCol</a> <font color="Maroon" title="c5">AD</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">pD</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</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="c5">AD</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">AD</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</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="c5">AD</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
<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> <font color="Maroon" title="c5">AD</font>
 ;<br/><a NAME="E2:21_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">AD</font></span>)</span>
 <b>by </b><i><a class="txt" href="laplace.html#E1:21_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#NK4" title="LAPLACE:NK.4">RCol</a> <font color="Maroon" title="c5">AD</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">pD</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">AD</font> <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>
 <b>by </b><i><a class="txt" href="laplace.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="laplace.html#E1:21_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="laplace.html#E1:21_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="laplace.html#D2" target="_self" title="LAPLACE:def.2">DEF1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_1"/>
 <a href="laplace.html#NK4" title="LAPLACE:NK.4">RCol</a> <font color="Maroon" title="c5">AD</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">AD</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">AD</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2"/>
 <a href="laplace.html#NK4" title="LAPLACE:NK.4">RCol</a> <font color="Maroon" title="c5">AD</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">pD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">AD</font>
 <b>by </b><i><a class="ref" href="laplace.html#D2" target="_self" title="LAPLACE:def.2">DEF1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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