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

<b>thus </b><a NAME="E1:18_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font> <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="c5">M</font> implies  ex <font color="Olive" title="b1">M1</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">D</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M1</font> <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="c5">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">M1</font> <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="c5">M</font> &amp; ( for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M</font> holds <br/>( ( <font color="Olive" title="b3">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> implies <font color="Olive" title="b1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> ) &amp; ( <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> implies <font color="Olive" title="b1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font> <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="c5">M</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>( ( <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) );<br/>
<b>reconsider </b><font color="Maroon" title="c7">V</font> = <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c8">U</font> = <font color="Maroon" title="c3">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:18_1_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c7">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c8">U</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> holds <br/>for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">x1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">x2</font>] holds <br/><font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c7">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c8">U</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font>;<br/>

<b>assume </b><a NAME="E2:18_1_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">x1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c12">x2</font>] )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c10">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:18_1_1_1_1"/>
( ( <font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font> ) &amp; ( <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> ) &amp; ( <font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font> ) &amp; ( <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> ) )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:18_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_1_1_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:18_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c7">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c8">U</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> holds <br/> ex <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c7">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c8">U</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:18_1_1_1_2_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:18_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font></span>)</span>
 <b>by </b><i><a class="txt" href="laplace.html#E1:18_1_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="laplace.html#E1:18_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:18_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">pD</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:18_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</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="c6">pD</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">pD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E6:18_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>] )
 <b>by </b><i><a class="txt" href="laplace.html#E1:18_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E7:18_1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">x</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:18_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>]
 ;<br/><b>hence </b><a NAME="E3:18_1_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">x</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">M1</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c8">U</font>,<font color="Maroon" title="c4">D</font><b> such that </b><br/><a NAME="E6:18_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c9">M1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c9">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 <b>from </b><i><a class="ref" href="matrix_1.html#S2" title="MATRIX_1:sch.2">MATRIX_1:sch 2</a>(<a class="txt" href="laplace.html#E3:18_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="laplace.html#E4:18_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">M1</font> = <font color="Maroon" title="c9">M1</font> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">D</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">M1</font>
;<br/>

<div><i><font color="Green" title="E21">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_3_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_3_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E22">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:18_1_1_1_3_1_1"/><b>then </b>
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; 0 <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="c5">M</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <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="c10">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">M1</font> <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="c5">M</font> )
 <b>by </b><i><a class="txt" href="laplace.html#E2:18_1_1_1_3_1_1"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_3_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_3_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_1_3_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <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="c10">M1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font> <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="c10">M1</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c5">M</font> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c4">D</font> <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<a NAME="E10:18_1_1_1"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c10">M1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/>
<b>hence </b><a NAME="E11:18_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">M1</font> <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="c5">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c10">M1</font> <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="c5">M</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M</font> holds <br/>( ( <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c10">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) &amp; ( <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font> implies <font color="Maroon" title="c10">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) )
 <b>by </b><i><a class="txt" href="laplace.html#E6:18_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="laplace.html#E8:18_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:18_1_1"/>
( not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">pD</font> <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="c5">M</font> implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">D</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> )
 ;<br/>


</div>
