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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c2">K</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E6">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> )
 ;<br/>

<a NAME="E2:17"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixr1.html#T16" target="_self" title="MATRIXR1:th.16">Th16</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E8">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</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="c4">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</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="c4">M</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:17"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:17"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="matrixr1.html#E3:17"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green" title="E11">A6</font></i>: 
 <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"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E4:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:17"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1"/>
<font color="Maroon" title="c1">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="c4">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E1:17"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c1">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="c4">M</font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E7:17"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="matrixr1.html#E1:17_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:17_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E7:17"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="matrixr1.html#E1:17_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="matrixr1.html#E5:17"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="matrixr1.html#E6:17"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="matrixr1.html#E1:17_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="matrixr1.html#E4:17_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:17"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c4">M</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixr1.html#E2:17"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="matrixr1.html#E3:17"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="matrixr1.html#E4:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>


</div>
