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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:39_1_1_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1_1_1"/>
<font color="Maroon">M1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">M2</font> is    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:39_1_1_1_2"/><b>then </b>
<font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1_1_2"/>
<font color="Maroon">M1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">M2</font> is    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:39_1_1_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E3:39_1_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:39_1_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:39_1_1_1_3"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><a NAME="E6:39_1_1_1_3"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><a NAME="E7:39_1_1_1_3"/><i><font color="Green">E65</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E8:39_1_1_1_3"/><i><font color="Green">E66</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E9:39_1_1_1_3"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>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">M1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M2</font></span>)</span> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">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">M1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1_1_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<a NAME="E4:39_1_1_1_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_3_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_3_1_1_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E5:39_1_1_1_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_3_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_3_1_1_1_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E5:39_1_1_1_3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">y2</font>
 <b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_3_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_1_1_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M3</font> = <font color="Maroon">M1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M2</font> as   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/><a NAME="E11:39_1_1_1_3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><a NAME="E12:39_1_1_1_3"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E13:39_1_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><a NAME="E14:39_1_1_1_3"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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="E16:39_1_1_1_3"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M3</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M3</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><a NAME="E17:39_1_1_1_3"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_3_2_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font> or <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_3_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_3_2_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E5:39_1_1_1_3_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_3_2_1_1"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_3_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_3_2_1_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E5:39_1_1_1_3_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M2</font>,<font color="Maroon">y2</font>
 <b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_3_2_1_2"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:39_1_1_1_3"/>
<font color="Maroon">M1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">M2</font> is    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
