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

<b>let </b><font color="Maroon">D</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green">E49</font></i>: 
not  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> );<br/>
<a NAME="E3:81"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:81"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<i><font color="Green">E53</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
;<br/>
<a NAME="E3:81_1"/><i><font color="Green">E54</font></i>: 
0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>;<br/>
<a NAME="E4:81_1"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E5:81_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a>, </i>;<br/>
<a NAME="E6:81_1"/><b>then </b>
( 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T8" target="_self">Th8</a>, <a class="ref" href="card_4.html#T24" target="_self">Th24</a>, , <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>hence </b><a NAME="E7:81_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="card_4.html#T8" target="_self">Th8</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="txt" href="card_4.html#E4:81"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:81"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T13" target="_self">Th13</a>);<br/></i>
<a NAME="E6:81"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:81"/>
( <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>,<font color="Maroon">D</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
