<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">l</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Olive" title="b2">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font>  st <font color="Maroon" title="c1">l</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Olive" title="b1">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">l</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( <font color="Maroon" title="c1">l</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">k</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">and </span><br/><a NAME="E3:38"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</span></span><br/>

<a NAME="E4:38"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="matrix12.html#T13" target="_self" title="MATRIX12:th.13">Th13</a></span>;<br/>
<a NAME="E5:38"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E6:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:38"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#T10" title="MATRIX_6:th.10">MATRIX_6:10</a></span>;<br/>
<span class="kw">then </span> <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="matrix12.html#T17" target="_self" title="MATRIX12:th.17">Th17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="matrix12.html#T13" target="_self" title="MATRIX12:th.13">Th13</a>, <a class="ref" href="matrix_6.html#T13" title="MATRIX_6:th.13">MATRIX_6:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#T10" title="MATRIX_6:th.10">MATRIX_6:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="matrix12.html#T17" target="_self" title="MATRIX12:th.17">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:38"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a</font></span>)</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>
