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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">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="c6">A</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="c5">D</font>;<br/><b>let </b><font color="Maroon" title="c7">B</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="c3">k</font>,<font color="Maroon" title="c5">D</font>;<br/><b>let </b><font color="Maroon" title="c8">pA</font>, <font color="Maroon" title="c9">pB</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">D</font>;<br/>















<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E21">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">pA</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="c6">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">pB</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="c7">B</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c10">AB</font> = <font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font>;<br/>
<b>set </b><font color="Maroon" title="c11">RAB</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">RA</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font>;<br/>
<b>set </b><font color="Maroon" title="c13">RB</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font>;<br/>
<b>set </b><font color="Maroon" title="c14">Rab</font> = <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span>;<br/>
<a NAME="E2:21"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span></span>)</span> &amp;  <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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">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"/><i><font color="Green" title="E23">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><a NAME="E2:21_1"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:21_1"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c15">j</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> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:21_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/><a NAME="E4:21_1"/>
( <font color="Maroon" title="c8">pA</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c9">pB</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:21"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/><a NAME="E5:21_1"/><b>then </b>
<font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></i>;<br/><a NAME="E6:21_1"/><b>then </b><i><font color="Green" title="E25">A5</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="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E7:21_1"/><i><font color="Green" title="E26">A6</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="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:21_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><div><i><font color="Green" title="E27">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_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_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1"/><b>then </b>
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pA</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">pB</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:21"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="matrix15.html#E6:21_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="matrix15.html#E7:21_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_2"/><b>then </b>
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c15">j</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c15">j</font> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c15">j</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>;<br/><b>hence </b> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</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="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></i>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E8:21_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="matrix15.html#E3:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21"/>
 <a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">pB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c8">pA</font></span>)</span> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <span class="p1">(<span class="default"><a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c9">pB</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E2:21"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
