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<b>let </b><font color="Maroon" title="c1">W</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">W</font> is <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a>
 ;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<a NAME="E3:4"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:4_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="classes2.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="classes2.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></i>;<br/>
<a NAME="E5:4"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 
;<br/>
<a NAME="E6:4"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 
;<br/>
<b>hence </b><a NAME="E7:4"/>
( not <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">W</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> )
 <b>by </b><i><a class="txt" href="classes2.html#E4:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>


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