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<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">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:24"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</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> <font color="Maroon">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Olive">j</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">j</font>,<font color="Maroon">i</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">j</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> <font color="Maroon">M</font></span>)</span>
 ;<br/><a NAME="E2:24_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:24_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:24_1"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">M</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/><b>thus </b><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
<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="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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