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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">l</font>, <font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font> holds   <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c5">K</font> holds   <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span><br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c5">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span><br/>











<a NAME="E1:19"/><i><font color="Green" title="E15">A1</font></i>: 
(  <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="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">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="c6">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">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="c6">M</font> &amp;  <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="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</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="c6">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</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="c6">M</font> )
 
<b>by </b><i><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a>, <a class="ref" href="matrix12.html#T1" target="_self" title="MATRIX12:th.1">Th1</a></i>;<br/>
<a NAME="E2:19"/>
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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">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"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</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="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> )</span><br/>



<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E16">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</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="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><br/>

<a NAME="E2:19_1"/>
 <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="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c7">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="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix12.html#E1:19_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:19_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c4">k</font>,<font color="Maroon" title="c8">j</font> ) &amp; ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c3">l</font>,<font color="Maroon" title="c8">j</font> ) &amp; ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">l</font> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c7">i</font>,<font color="Maroon" title="c8">j</font> ) )
 
<b>by </b><i><a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>;<br/>
<a NAME="E5:19_1"/>
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c4">k</font>,<font color="Maroon" title="c8">j</font> ) &amp; ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c3">l</font>,<font color="Maroon" title="c8">j</font> ) &amp; ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">l</font> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font> implies <span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="c7">i</font>,<font color="Maroon" title="c8">j</font> ) )
 
<b>by </b><i><a class="txt" href="matrix12.html#E3:19_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="matrix12.html#D1" target="_self" title="MATRIX12:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:19_1"/>
<span class="p1">(<span class="default"><a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</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="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="matrix12.html#E4:19_1"><i><font color="Green" title="E18">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
 <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix12.html#NK4" title="MATRIX12:NK.4">ILine</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c3">l</font>
 <b>by </b><i><a class="txt" href="matrix12.html#E1:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
