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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">M2</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/>











<b>assume </b><a NAME="E1:45"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
 ;<br/>

<i><font color="Green">E44</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E50</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:45_3"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:45_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/><a NAME="E4:45_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:45_3"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:45_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:45_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:45_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/><a NAME="E9:45_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/><a NAME="E10:45_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">m2</font> = <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
<br/>.= 
 <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="matrix_2.html#T17">MATRIX_2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T17">MATRIX_2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_2.html#T17">MATRIX_2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">m2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
<br/>.= 
<font color="Maroon">m1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">m2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K4">@</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K9" target="_self">^^</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
