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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<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>assume </b><a NAME="E1:10"/><i><font color="Green">E52</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>
 ;<br/>

<a NAME="E2:10"/><i><font color="Green">E58</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E3:10"/><b>then </b><i><font color="Green">E59</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>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green">E61</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>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:10"/>
not <font color="Maroon">M</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:10"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E11:10"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <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#D4">MATRIX_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:10"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E14:10"/><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:10"/>
 ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:10"/>
(  not <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> ) )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


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