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

<b>let </b><font color="Maroon">X</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">F</font> be    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">I</font> be    <a href="card_fil.html#M2" target="_self">Ideal</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:16"/>
for <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or  not <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_1"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
 <a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
contradiction
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">I</font> )
 ;<br/>

<a NAME="E3:16"/><b>then </b>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T25">SUBSET_1:25</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
;<br/>
<b>hence </b><a NAME="E7:16"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
