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<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="E1:52"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1" target="_self">CARD_4:def 1</a><br/>

<a NAME="E2:52"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">1 <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#K3">alef</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:52"/><b>then </b><i><font color="Green">E44</font></i>: 
1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="card_4.html#V1" target="_self">countable</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="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="card_4.html#V1" target="_self">countable</a>;<br/>
<a NAME="E4:52"/>
<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> is <a href="card_4.html#V1" target="_self">countable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>;<br/>
<a NAME="E6:52"/><i><font color="Green">E50</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="52_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="E1:52_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:52_1"/><b>then </b>
( <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> is <a href="card_4.html#V1" target="_self">countable</a> &amp; <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>,<span class="p1">(<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> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="card_4.html#E1"><i><font color="Green">Lemma41</font></i></a>, , </i>;<br/>
<a NAME="E3:52_1"/><b>then </b>
(  <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> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <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> <a href="hidden.html#R1">=</a>  <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> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E4:52_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/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:52"/>
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#T1" target="_self">Th1</a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</a>);<br/></i>
<b>hence </b><a NAME="E8:52"/>
<font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="card_4.html#V1" target="_self">countable</a>
 ;<br/>


</div>
