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

<b>let </b><font color="Maroon">M</font> be  non <a href="card_4.html#V1">countable</a> <a href="card_5.html#NM1">Aleph</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">M</font>
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<a NAME="E3:39"/>
for <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> &amp; <font color="Olive">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:39_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:39_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:39_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E6:39_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
 <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 <b>by </b><i/>;<br/>


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