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<b>let </b><font color="Maroon">A</font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>



<a NAME="E1:12"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:12"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a> iff  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:12"/>
(  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">A</font> iff (  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
(  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> iff  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<b>hence </b><a NAME="E6:12"/>
( not <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a> implies  ex <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B1</font> ) )
 <b>by </b><i>, </i>;<br/>

<b>given </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E7:12"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E8:12"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 ;<br/>

<a NAME="E9:12"/>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
not <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


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