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<b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">W</font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">W</font> is <a href="classes1.html#V1">subset-closed</a>
 
<b>by </b><i><a class="ref" href="classes1.html#D2">CLASSES1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E2:24_1"/>
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>
 ;<br/>

<a NAME="E2:24_1_1"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:24_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="classes1.html#T35">CLASSES1:35</a></i>;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="txt" href="classes2.html#E1"><i><font color="Green">Lemma25</font></i></a>, </i>;<br/>
<a NAME="E6:24_1_1"/><b>then </b>
(  <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 
<b>by </b><i>, , <a class="ref" href="classes1.html#T37">CLASSES1:37</a></i>;<br/>
<a NAME="E7:24_1_1"/><b>then </b>
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="classes2.html#E1"><i><font color="Green">Lemma25</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="classes1.html#T33">CLASSES1:33</a></i>;<br/>


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