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

<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>
<b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font>, <font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">n1</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">m1</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">d</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  ;<br/>
<a NAME="E5:15_1_1"/>
<font color="Maroon">M</font> is   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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> <font color="Maroon">M</font> 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="15_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">n1</font>
;<br/>

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

<b>assume </b><a NAME="E1:15_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:15_1_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">n1</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> ;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E4:15_1_1_1_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">n1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1_1_1"/>
<font color="Maroon">M</font> is   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font> ;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E7:15_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>

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

<b>assume </b><a NAME="E8:15_1_1"/><i><font color="Green">E57</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>
 ;<br/>

<a NAME="E9:15_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E10:15_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:15_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><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>


</div>
