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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is   <a href="card_1.html#NM1">Cardinal</a>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22_3_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">N</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="card_3.html#NM1">Cardinal-Function</a><b> such that </b><br/><a NAME="E4:22_3_1"/><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="card_3.html#S1">CARD_3:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E5:22_3_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">x</font> is   <a href="card_1.html#NM1">Cardinal</a> ) )
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">K</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><a NAME="E6:22_3_1"/>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E7:22_3_1"/><b>then </b>
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:22_3_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">K</font>,<font color="Maroon">N</font>
 <b>by </b><i/>;<br/>


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