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

<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>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><span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</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">a<sub>1</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0;<br/>
<a NAME="E1:63"/>
( <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K3">alef</a> 0 )
 
<b>by </b><i>, <a class="ref" href="card_2.html#T32">CARD_2:32</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:63"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:63"/><i><font color="Green">E49</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="63_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="E1:63_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:63_1"/><i><font color="Green">E51</font></i>: 
not  <a href="card_1.html#K3">alef</a> 0 is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
;<br/>
<a NAME="E4:63_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <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="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">k</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <a href="ordinal1.html#K5">omega</a> </span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="card_2.html#T25">CARD_2:25</a></i>
<br/>.= 
 <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="ordinal2.html#K11">*^</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></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="ordinal2.html#K11">*^</a> <a href="ordinal1.html#K5">omega</a> </span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="card_2.html#T24">CARD_2:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="card_2.html#T25">CARD_2:25</a></i>
<br/>.= 
 <a href="card_1.html#K3">alef</a> 0
<b>by </b><i><a class="ref" href="card_4.html#T6" target="_self">Th6</a>, , </i>
;<br/>
<b>hence </b><a NAME="E6:63_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:63"/><i><font color="Green">E52</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>  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#T5" target="_self">Th5</a>);<br/></i>
<b>hence </b><a NAME="E5:63"/>
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</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">n</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 ;<br/>

<b>thus </b><a NAME="E6:63"/>
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">n</font></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</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><a class="ref" href="card_4.html#T6" target="_self">Th6</a></i>;<br/>


</div>
