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<span class="kw">let </span><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</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><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="c3">n</font>,<font color="Maroon" title="c4">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="Maroon" title="c2">k</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="Maroon" title="c3">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="c4">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Olive" title="b4">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Olive" title="b3">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="c5">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="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">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="Maroon" title="c2">k</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="Maroon" title="c3">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="c4">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Olive" title="b3">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">K</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">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="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">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="c2">k</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="c3">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="c4">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Olive" title="b2">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c8">M1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">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#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">M</font></span>)</span> &amp; <font color="Maroon" title="c2">k</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">M</font></span>)</span> &amp; <font color="Maroon" title="c3">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="c4">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="c8">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><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#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">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c2">k</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">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c4">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> 
 <span class="kw">and </span><br/><a NAME="E5:33"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c8">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span><br/>

<a NAME="E6:33"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E7:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<a NAME="E8:33"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></span>;<br/>
<a NAME="E9:33"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E10:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c8">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<a NAME="E11:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 <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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<span class="lab"><font color="Green" title="E36">A11</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">M1</font> = 
 <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="c8">M1</font></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 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">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>
;<br/>
<a NAME="E13:33"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="matrix12.html#D3" target="_self" title="MATRIX12:def.3">Def3</a></span>;<br/>
<a NAME="E14:33"/><span class="kw">then </span>
 <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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<a NAME="E15:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">M2</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">K</font><span class="kw"> such that </span><br/><a NAME="E17:33"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c9">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ;<br/>
<a NAME="E18:33"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
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="c7">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="c7">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="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c2">k</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="c7">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="c9">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="c7">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="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">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"> ( <font color="Maroon" title="c10">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="c7">M</font> &amp; <font color="Maroon" title="c11">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="c7">M</font></span>)</span> implies ( ( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">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="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33_2"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c10">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="c7">M</font>
 <span class="kw">and </span><br/><a NAME="E2:33_2"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<font color="Maroon" title="c11">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="c7">M</font></span>)</span>
 ; <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="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> ) &amp; ( <font color="Maroon" title="c11">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="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> ) )</span><br/>

<a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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="c7">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E2:33_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c10">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="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E5:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<font color="Maroon" title="c10">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="c8">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E6:33_2"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M1</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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E7:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c2">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> <font color="Maroon" title="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</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 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="c8">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="matrix12.html#D3" target="_self" title="MATRIX12:def.3">Def3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">M1</font>
<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="E9:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c10">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></span>;<br/>
<a NAME="E10:33_2"/><span class="lab"><font color="Green" title="E48">A23</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> <font color="Maroon" title="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:33_2"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> )
  <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="c11">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="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_2_2"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:33_2_2"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c11">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span></span><br/>

<font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E9:33_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c10">i</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="c8">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="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_2"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="txt" href="#E7:33_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E2:33_2_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="matrix12.html#D3" target="_self" title="MATRIX12:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c8">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="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:33_2_2"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E2:33_2_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:33_2_2"/>
<font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</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><span class="kw">end;</span></div>

<a NAME="E12:33_2"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
<font color="Maroon" title="c11">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="c8">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:33_2"/>
( <font color="Maroon" title="c11">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="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_2_3"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c11">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><br/>

<font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> = 
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E9:33_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c8">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E12:33_2"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E1:33_2_3"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="matrix12.html#D3" target="_self" title="MATRIX12:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:33_2_3"/>
<font color="Maroon" title="c9">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
 ; <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>
<a NAME="E19:33"/>
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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="33_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:33_3"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:33_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font></span><br/>

<a NAME="E3:33_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A29</font></span>: 
( <font color="Maroon" title="c10">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="c7">M</font> &amp; <font color="Maroon" title="c11">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="c7">M</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_3"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:33_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A30</font></span>: 
( <font color="Maroon" title="c11">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E39">A14</font></span></a></span>;<br/>
<a NAME="E5:33_3"/><span class="lab"><font color="Green" title="E43">A31</font></span>: 
( <font color="Maroon" title="c11">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E41">A29</font></span></a></span>;<br/>
<a NAME="E6:33_3"/>
( <font color="Maroon" title="c11">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c2">k</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="c7">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c1">l</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="matrix12.html#D6" target="_self" title="MATRIX12:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:33_3"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E4:33_3"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E5:33_3"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="matrix12.html#D6" target="_self" title="MATRIX12:def.6">Def6</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>
<span class="kw">hence </span><a NAME="E20:33"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK6" title="MATRIX12:NK.6">RLineXS</a> <font color="Maroon" title="c8">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">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#NK12" title="MATRIX12:NK.12">RColXS</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</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>
