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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being 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/> for <font color="Olive" title="b2">A</font> being   <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="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">B</font> being   <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="Olive" title="b1">D</font><br/> for <font color="Olive" title="b4">pA</font>, <font color="Olive" title="b5">pB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">D</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b3">B</font> holds <br/> <a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Olive" title="b3">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b5">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="Olive" title="b2">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b4">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="Olive" title="b3">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b5">pB</font></span>)</span></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <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/> for <font color="Olive" title="b2">B</font> being   <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/> for <font color="Olive" title="b3">pA</font>, <font color="Olive" title="b4">pB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">D</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b1">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b2">B</font> holds <br/> <a href="matrix11.html#K3" title="MATRIX11:func.3">ReplaceLine</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix15.html#K1" title="MATRIX15:func.1">^^</a> <font color="Olive" title="b2">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">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="Olive" title="b1">A</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b3">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="Olive" title="b2">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b4">pB</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being   <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/> for <font color="Olive" title="b2">pA</font>, <font color="Olive" title="b3">pB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">D</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b3">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="Olive" title="b1">B</font> holds <br/> <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="Olive" title="b1">B</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">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="Olive" title="b2">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="Olive" title="b1">B</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b3">pB</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">pA</font>, <font color="Olive" title="b2">pB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">D</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">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> holds <br/> <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="Olive" title="b1">pA</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">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="Olive" title="b1">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="Olive" title="b2">pB</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> implies  <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> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">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/>
<span class="kw">set </span><font color="Maroon" title="c11">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/>
<span class="kw">set </span><font color="Maroon" title="c12">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/>
<span class="kw">set </span><font color="Maroon" title="c13">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/>
<span class="kw">set </span><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/>
<div><span class="lab"><font color="Green" title="E23">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/><a NAME="E2:21_1"/><span class="kw">then </span>
<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#NM4" title="FINSEQ_2:NM.4">Tuple</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="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>,<font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>;<br/><a NAME="E3:21_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T151" title="FINSEQ_2:th.151">FINSEQ_2:151</a></span>;<br/><a NAME="E4:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">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="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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c15">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="Red">::  thesis: </font></span></a><span class="hide"> ( 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> implies <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="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )</span><br/><span class="kw">assume </span><a NAME="E5:21_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span><br/><a NAME="E6:21_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E7:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a></span>;<br/><a NAME="E8:21_1"/><span class="lab"><font color="Green" title="E27">A7</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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/><div><span class="lab"><font color="Green" title="E28">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><a NAME="E2:21_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></span>;<br/><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> <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E8:21_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E1:21_1_1_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E2:21_1_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><a NAME="E2:21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A12</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></span>;<br/><a NAME="E3:21_1_1_1_2"/>
(  <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E1:21_1_1_1_2"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></span>;<br/><span class="kw">hence </span> <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="binarith.html#K9" title="BINARITH:func.9">^</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E2:21_1_1_1_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="matrix15.html#T15" target="_self" title="MATRIX15:th.15">Th15</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E9:21_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:21"/>
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
