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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">I</font> be    <a href="card_fil.html#M2" title="CARD_FIL:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>thus </b><a NAME="E1:16"/>
for <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or  not <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">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" title="c4">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>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">F</font> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:16_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>
<a NAME="E4:16_1"/>
<font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:16_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">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>assume </b><a NAME="E2:16"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">I</font> )
 ;<br/>

<a NAME="E3:16"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E4:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:16"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="card_fil.html#D2" target="_self" title="CARD_FIL:def.2">Def2</a></i>;<br/>
<a NAME="E5:16"/><b>then </b>
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T25" title="SUBSET_1:th.25">SUBSET_1:25</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 
;<br/>
<b>hence </b><a NAME="E7:16"/>
contradiction
 <b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self" title="CARD_FIL:def.2">Def2</a></i>;<br/>


</div>
