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



<b>assume </b><a NAME="E1:36"/>
<font color="Maroon">W</font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E27</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/>
<b>assume </b><a NAME="E3:36"/>
<font color="Maroon">X</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 ;<br/>

<a NAME="E4:36"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:36"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E6:36"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:36"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T59">CARD_3:59</a></i>;<br/>
<a NAME="E9:36"/><b>then </b>
(  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font></span>)</span> <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> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T42">CLASSES1:42</a>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:36"/>
<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>
 <b>by </b><i><a class="ref" href="classes1.html#T47">CLASSES1:47</a></i>;<br/>


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