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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:28_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
;<br/>
<b>thus </b><a NAME="E3:28_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E2:28_1"><i><font color="Green">E21</font></i></a>);</i><br/>


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

<b>assume </b><a NAME="E2:28"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:28"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:28"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
