<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>thus </b><a NAME="E1:33"/>
( <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <font color="Maroon" title="c1">X</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> $1</span>)</span> <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">X</font>;<br/>
<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:33_1"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y1</font> where <font color="Olive" title="b1">Y1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y1</font>] </span>}</span> 
 
;<br/>
<b>thus </b><a NAME="E3:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">Y</font>]
 <b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E2:33_1"><i><font color="Green" title="E24">A1</font></i></a>);</i><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:33"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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">X</font> implies <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>


</div>
