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<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  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> iff <font color="Olive">Y</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:9"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:9"/><i><font color="Green">E24</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:9"/>
not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">Y1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Y2</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E6:9"/>
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> implies <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/>
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
( <font color="Maroon">Y1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y2</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:9_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font></span>)</span> <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="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E7:9"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> )
 ;<br/>

<a NAME="E8:9"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y1</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:9"/>
<font color="Maroon">Y1</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon">Y2</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:9"/>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div>
