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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">K</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>





<a NAME="E1:44"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c3">A</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7" target="_self" title="MATRIX_1:def.7">Def7</a></i>;<br/>
<a NAME="E2:44"/><i><font color="Green" title="E27">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" target="_self" title="MATRIX_1:def.3">Def3</a></i>;<br/>
<div><i><font color="Green" title="E28">A3</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" title="E29">A4</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="matrix_1.html#E1:44_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:44"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" target="_self" title="MATRIX_1:th.20">Th20</a>, <a class="txt" href="matrix_1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a></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" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:44"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="matrix_1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="matrix_1.html#D4" target="_self" title="MATRIX_1:def.4">Def4</a></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" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">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" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">K</font>;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">n2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:44_2"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> holds <br/> ex <font color="Olive" title="b2">s</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n2</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1" target="_self" title="MATRIX_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:44_2"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">p2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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" title="c6">p2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">K</font>;<br/>

<b>assume </b><a NAME="E1:44_2_1"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n2</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E3:44_2"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:44_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n2</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E3:44_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E6:44_2"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n2</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:44_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="matrix_1.html#E3:44_2"><i><font color="Green" title="E30">A6</font></i></a></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" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E33">A10</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T20" target="_self" title="MATRIX_1:th.20">Th20</a>, <a class="txt" href="matrix_1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/><a NAME="E3:44_2_2_1_1"/><b>then </b><i><font color="Green" title="E35">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c3">A</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c3">A</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:44_2_2_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="matrix_1.html#E37"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/><a NAME="E4:44_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E4:44_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="matrix_1.html#D3" target="_self" title="MATRIX_1:def.3">Def3</a>, <a class="txt" href="matrix_1.html#E2:44_2_2_1_1"><i><font color="Green" title="E34">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:44_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E6:44_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="matrix_1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="matrix_1.html#E3:44_2_2_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="matrix_1.html#E2:44_2_2_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="matrix_1.html#E1:44_2_2_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="matrix_1.html#T20" target="_self" title="MATRIX_1:th.20">Th20</a></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" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_2"/><b>then </b><i><font color="Green" title="E33">A13</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" target="_self" title="MATRIX_1:def.4">Def4</a>, <a class="txt" href="matrix_1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/><a NAME="E3:44_2_2_1_2"/><i><font color="Green" title="E34">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c3">A</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D7" target="_self" title="MATRIX_1:def.7">Def7</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_2_2_1_2"/><i><font color="Green" title="E35">A15</font></i>: 
( <font color="Maroon" title="c7">x0</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:44_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E6:44_2_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E2:44_2_2_1_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="matrix_1.html#E3:44_2_2_1_2"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="matrix_1.html#E5:44_2_2_1_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:44"/>
<font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" target="_self" title="MATRIX_1:def.3">Def3</a>, <a class="txt" href="matrix_1.html#E3:44"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>


</div>
