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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">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="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="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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b3">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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">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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b2">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span><br/><span class="kw">let </span><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="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="c6">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="c6">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="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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E24">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="c6">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E25">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="c6">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E26">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="E4:31"/><span class="lab"><font color="Green" title="E27">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="E5:31"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span><br/>

<a NAME="E6:31"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 <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="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">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="E7:31"/><span class="lab"><font color="Green" title="E30">A7</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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>
 
<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="E8:31"/>
 <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="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">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="E9:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
 <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="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E29">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="E10:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A9</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<a NAME="E11:31"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></span>;<br/>
<a NAME="E12:31"/>
 <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="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E29">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="E13:31"/><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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
<a NAME="E14:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E32">A9</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="c8">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="E16:31"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 ;<br/>
<a NAME="E17:31"/><span class="lab"><font color="Green" title="E36">A13</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="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="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="c2">k</font> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</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="Maroon" title="c1">l</font> ) &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> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="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="c2">k</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</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="c1">l</font> ) &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> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:31_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="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="c2">k</font> ) &amp; ( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</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="c1">l</font> ) &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> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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="E3:31_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">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="E5:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: 
( <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> )
 
<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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="c7">M1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">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="E7:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:31_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="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="c2">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="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</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="c1">l</font> ) &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> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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="31_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1_2"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<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="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="c6">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E37">A14</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="E2:31_1_2"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E42">A19</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="c7">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="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E2:31_1_2"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></span>
<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="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:31_1_2"><span class="lab"><font color="Green" title="E43">A20</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="E4:31_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="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="c2">k</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>

<span class="kw">thus </span><a NAME="E9:31_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</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="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="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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="31_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1_3"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
<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="c1">l</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E37">A14</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="E2:31_1_3"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="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="c1">l</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E42">A19</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="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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E2:31_1_3"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></span>
<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="c1">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:31_1_3"><span class="lab"><font color="Green" title="E43">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="kw">hence </span><a NAME="E4:31_1_3"/>
<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="c1">l</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>

<span class="kw">assume </span><a NAME="E10:31_1"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
( <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> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">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="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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E42">A19</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="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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E10:31_1"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E39">A16</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="E12:31_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:()"><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="E18:31"/>
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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="31_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
 <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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="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:()"><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="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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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/>



<span class="kw">assume </span><a NAME="E2:31_2"/>
<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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="E3:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
( <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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="c2">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E17:31"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>;<br/>
<a NAME="E5:31_2"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="c1">l</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="matrix12.html#D4" target="_self" title="MATRIX12:def.4">Def4</a></span>;<br/>
<a NAME="E6:31_2"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="c1">l</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E17:31"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a></span>;<br/>
<a NAME="E7:31_2"/><span class="lab"><font color="Green" title="E42">A30</font></span>: 
( <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> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:31"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E17:31"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a></span>;<br/>
<a NAME="E8:31_2"/>
( <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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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="c2">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="matrix12.html#D4" target="_self" title="MATRIX12:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:31_2"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E6:31_2"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E5:31_2"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="matrix12.html#D4" target="_self" title="MATRIX12:def.4">Def4</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="E19:31"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</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#NK10" title="MATRIX12:NK.10">ICol</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c1">l</font>,<font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">A11</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>
