<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b3">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> holds <br/><font color="Olive" title="b3">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Z</font> ) holds <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b1">Z</font> &amp;  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b1">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b2">F</font>,<font color="Olive" title="b1">Z</font> )</span><br/><b>let </b><font color="Maroon" title="c2">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b2">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Z</font> ) holds <br/>( <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">Z</font> &amp;  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">Z</font> )</span><br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Olive" title="b1">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Z</font> ) implies ( <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp;  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> ) )</span><br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Olive" title="b1">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp;  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )</span><br/>

<b>thus </b><a NAME="E2:26"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )</span><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" title="b1">Z2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">Z2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="ref" href="card_fil.html#T5" target="_self" title="CARD_FIL:th.5">Th5</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1_1"/>
 ex <font color="Olive" title="b1">Z2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">Z2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:26"/>
 <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span><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" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">CARD_FIL:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y1</font>, <font color="Olive" title="b2">Y2</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/> ( ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> implies <font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> ) &amp; ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Olive" title="b1">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y2</font> implies <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> ) )</span><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" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">Z2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:26_2_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c4">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E4:26_2_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">Z2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>;<br/>
<a NAME="E5:26_2_1"/>
<font color="Maroon" title="c4">Z2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="card_fil.html#E3:26_2_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E6:26_2_1"/><b>then </b>
<font color="Maroon" title="c4">Z2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><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="E7:26_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E4:26_2_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> ) &amp; ( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> ) )</span><br/>



<b>thus </b><a NAME="E2:26_2"/>
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> implies <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )</span><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" title="E21">A4</font></i>: 
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">Y3</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:26_2_2"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E4:26_2_2"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y1</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">Y4</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:26_2_2"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c7">Y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E7:26_2_2"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c7">Y4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:26_2_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>;<br/>
<a NAME="E8:26_2_2"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:26_2_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="card_fil.html#E6:26_2_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Y4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> = 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Y4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Y4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Y4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E10:26_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Y4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:26_2_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="card_fil.html#E7:26_2_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E11:26_2_2"/>
<font color="Maroon" title="c4">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E8:26_2_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:26_2"/>
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E21">A10</font></i>: 
( <font color="Maroon" title="c4">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">Y3</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:26_2_3"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E4:26_2_3"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y1</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>;<br/>
<a NAME="E5:26_2_3"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26_2_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="card_fil.html#E4:26_2_3"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:26_2_3"/>
<font color="Maroon" title="c6">Y3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">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" title="E25">A14</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E2:26_2_3_1"/>
<font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E3:26_2_3_1"/><b>then </b><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c6">Y3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26_2_3_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E4:26_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Y3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:26_2_3"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:26_2_3_1"/>
<font color="Maroon" title="c6">Y3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fil.html#E3:26_2_3_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:26_2_3"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:26_2_3"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>
<a NAME="E8:26_2_3"/><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="card_fil.html#E8:26_2_3"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font>

;<br/>
<a NAME="E10:26_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E11:26_2_3"/>
<font color="Maroon" title="c5">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E7:26_2_3"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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

<b>thus </b><a NAME="E4:26"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green" title="E21">A18</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">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:26_3"/>
( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:26_3"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:26_3"/>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self" title="CARD_FIL:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
