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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</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">A</font> be   <a href="matrix_1.html#NM3">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>





<a NAME="E1:44"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">K</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><a NAME="E1:44_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>case </b></a><a NAME="E1:44_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><b>case </b></a><a NAME="E1:44_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:44"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">K</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E1:44_2"/><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> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">n2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:44_2"/><i><font color="Green">E63</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> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> 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">n2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:44_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">p2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">K</font>  st <font color="Olive">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E1:44_2_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:44_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:44_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1"><b>case </b></a><a NAME="E1:44_2_2_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:44_2_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:44_2_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a>  is    <a href="matrix_1.html#M1" target="_self">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">n2</font>,<font color="Maroon">K</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:44_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_2"><b>case </b></a><a NAME="E1:44_2_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:44_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E4:44_2_2_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">x0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:44_2_2_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">x0</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">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E7:44_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:44_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:44"/>
<font color="Maroon">A</font> <a href="matrix_1.html#K4" target="_self">@</a>  is   <a href="matrix_1.html#NM3">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>
 <b>by </b><i>, </i>;<br/>


</div>
