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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">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<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">Z1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>







<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font> implies  ex <font color="Olive">Z2</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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> ] <b>means </b> ex <font color="Olive">Y2</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> where <font color="Olive">Y1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  &amp; <font color="Olive">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] </span>}</span> 
 
;<br/>
<a NAME="E3:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Z1</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self">CARD_FIL:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">Y2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> where <font color="Olive">Y1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:25_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 ;<br/>
<b>consider </b><font color="Maroon">Y3</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:25_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E9:25_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Y3</font>
;<br/>

<b>thus </b><a NAME="E10:25_1"/>
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:25_1"/>
<font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, <a class="txt" href="card_fil.html#E2:25_1"><i><font color="Green">E22</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">Z2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E2:25"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 ;<br/>

<a NAME="E4:25"/>
 ex <font color="Olive">Y2</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> where <font color="Olive">Y1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  &amp; <font color="Olive">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>
;<br/>

<b>thus </b><a NAME="E1:25_2"/>
<font color="Maroon">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> where <font color="Olive">Y1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:25_2"/>
<font color="Maroon">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25"/><b>then </b>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Y2</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> where <font color="Olive">Y1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  &amp; <font color="Olive">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:25"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font>
 ;<br/>


</div>
