<?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">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>assume </b><a NAME="E1:26"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">Y1</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font>
 ;<br/>

<b>thus </b><a NAME="E2:26"/>
<font color="Maroon">Z</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1"/>
 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">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1_1"/>
 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">Z</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">Z</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>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:26"/>
 <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_2"/>
not  <a href="xboole_0.html#K1">{}</a>  <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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D1" target="_self">CARD_FIL:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  <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/>

<b>then consider </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><br/><a NAME="E3:26_2_1"/><i><font color="Green">E24</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="E4:26_2_1"/><i><font color="Green">E25</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>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>
<a NAME="E5:26_2_1"/>
<font color="Maroon">Z2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:26_2_1"/><b>then </b>
<font color="Maroon">Z2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:26_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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

<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="E2:26_2"/>
( <font color="Maroon">Y1</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> &amp; <font color="Maroon">Y2</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 <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y2</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Y1</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> &amp; <font color="Maroon">Y2</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/>

<b>then 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="E3:26_2_2"/><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>and </b><br/><a NAME="E4:26_2_2"/><i><font color="Green">E59</font></i>: 
<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">Y1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y4</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:26_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:26_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">Y4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:26_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26"><i><font color="Green">E22</font></i></a>, , </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> = 
<font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">Y4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E10:26_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E11:26_2_2"/>
<font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y2</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>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E3:26_2"/>
( <font color="Maroon">Y1</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> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> implies <font color="Maroon">Y2</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">Y1</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> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> )
 ;<br/>

<b>then 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="E3:26_2_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:26_2_3"/><i><font color="Green">E71</font></i>: 
<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">Y1</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:26_2_3"/><i><font color="Green">E72</font></i>: 
<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">Y2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:26_2_3"/>
<font color="Maroon">Y3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_2_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E2:26_2_3_1"/>
<font color="Maroon">Y3</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E3:26_2_3_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">Y3</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E4:26_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Y3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:26_2_3_1"/>
<font color="Maroon">Y3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E8:26_2_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon">Y2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font>

;<br/>
<a NAME="E10:26_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E11:26_2_3"/>
<font color="Maroon">Y2</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>
 <b>by </b><i>, </i>;<br/>


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


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

<b>thus </b><a NAME="E4:26"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">F</font>,<font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:26_3"/>
( <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="subset_1.html#K9">/\</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3"/>
<font color="Maroon">Y</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>
 <b>by </b><i/>;<br/>


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


</div>
