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<b>set </b><font color="Maroon" title="c8">pK</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>;<br/>
<b>set </b><font color="Maroon" title="c9">qK</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font>;<br/>
<b>set </b><font color="Maroon" title="c10">M1</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:10_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
 <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"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></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="c4">M</font>
 
<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="E2:10_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a></i>;<br/>
<a NAME="E3:10_1_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></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="c4">M</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> implies <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">l</font>,<font color="Olive" title="b2">j</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l</font> implies <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) ) ) )</span><br/>

<b>thus </b><a NAME="E4:10_1_1"/>
(  <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> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></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="c4">M</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> implies <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,<font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">l</font>,<font color="Olive" title="b2">j</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">l</font> implies <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) ) ) )
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E1:10_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="matrix12.html#E9"><i><font color="Green" title="E9">Lm3</font></i></a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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