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<b>let </b><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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><br/> for <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">M1</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<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#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="Olive" title="b3">M</font></span>)</span> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</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>  &amp; <font color="Maroon" title="c3">m</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>  &amp; <font color="Olive" title="b4">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Olive" title="b4">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b2">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Olive" title="b3">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b2">a</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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><br/> for <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">M1</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<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#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="Olive" title="b2">M</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</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>  &amp; <font color="Maroon" title="c3">m</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>  &amp; <font color="Olive" title="b3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Olive" title="b3">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b1">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Olive" title="b1">a</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">M1</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<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#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="Olive" title="b1">M</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</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>  &amp; <font color="Maroon" title="c3">m</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>  &amp; <font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Olive" title="b2">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span><br/><b>let </b><font color="Maroon" title="c6">M</font>, <font color="Maroon" title="c7">M1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#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="c6">M</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</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>  &amp; <font color="Maroon" title="c3">m</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>  &amp; <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  implies <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font> )</span><br/>













<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">l</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="c6">M</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</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>  &amp; <font color="Maroon" title="c3">m</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>  &amp; <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span><br/>

<a NAME="E2:32"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> <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> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E4:32"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></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="c7">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></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="c7">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></i>;<br/>
<a NAME="E5:32"/><b>then </b>
(  <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="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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"><span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E3:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">K</font>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">M2</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">K</font><b> such that </b><br/><a NAME="E8:32"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c8">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ;<br/>
<a NAME="E9:32"/><i><font color="Green" title="E29">A5</font></i>: 
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="c6">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="c6">M</font></span>)</span> holds <br/>( ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">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="c6">M</font> &amp; <font color="Maroon" title="c10">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="c6">M</font></span>)</span> implies ( ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:32_1"/>
( <font color="Maroon" title="c9">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="c6">M</font> &amp; <font color="Maroon" title="c10">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="c6">M</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) )</span><br/>

<a NAME="E2:32_1"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:32_1"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M1</font>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c10">j</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="c7">M1</font> &amp; <font color="Maroon" title="c9">i</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="c7">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> = 
 <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="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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> <font color="Maroon" title="c7">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">M1</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/>
<a NAME="E6:32_1"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:32_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E7:32_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_2"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span></span><br/>

<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="#E8:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E6:32_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E1:32_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E2:32_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E1:32_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E3:32_1_2"/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E8:32_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><br/>

<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="#E8:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E6:32_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E8:32_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="matrix12.html#D2" target="_self" title="MATRIX12:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E2:32_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_1"/>
<font color="Maroon" title="c8">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:32"/>
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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></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"><span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">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="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )</span><br/>



<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E30">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">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="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><br/>

<a NAME="E2:32_2"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E8:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/>
<a NAME="E3:32_2"/><b>then </b>
( <font color="Maroon" title="c9">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="c6">M</font> &amp; <font color="Maroon" title="c10">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="c6">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:32_2"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b>
( ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E8:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E9:32"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="matrix12.html#D5" target="_self" title="MATRIX12:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:32_2"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">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="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:32"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK5" title="MATRIX12:NK.5">SXLine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK11" title="MATRIX12:NK.11">SXCol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="#E6:32"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
