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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 );<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Z</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">D</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E5:26_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E6:26_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E7:26_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c3">X</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="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="filter_0.html#E5:26_1_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Z</font> be    <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"> ( <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> )</span><br/><b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font></span><br/><a NAME="E2:26_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 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>then reconsider </b><font color="Maroon" title="c6">F</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">X</font> as  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>  <b>by </b><i><a class="txt" href="filter_0.html#E7:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E10:26_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>thus </b><a NAME="E1:26_1_1_2"/>
(  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E6:26_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 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>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E12:26_1_1"/>
<font color="Maroon" title="c7">F</font> is    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">FILTER_0: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">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> ) iff <font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )</span><br/><b>let </b><font color="Maroon" title="c9">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> ) iff <font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )</span><br/>



<b>thus </b><a NAME="E1:26_1_1_3"/>
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> implies <font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> implies ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1_3_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font></span><br/>

<a NAME="E2:26_1_1_3_1"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">Z</font> be    <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"> ( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:26_1_1_3_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">Z'</font> = <font color="Maroon" title="c10">Z</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E3:26_1_1_3_1_1"/>
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z'</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z'</font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E1:26_1_1_3_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="filter_0.html#E1:26_1_1_3_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1_3_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">Def1</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>hence </b><a NAME="E3:26_1_1_3_1"/>
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E7:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 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>

<b>assume </b><a NAME="E2:26_1_1_3"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">Z</font> be    <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"> ( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/><b>assume </b><a NAME="E1:26_1_1_3_2"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">Z'</font> = <font color="Maroon" title="c10">Z</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><a NAME="E3:26_1_1_3_2"/>
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z'</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1_3"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="filter_0.html#E1:26_1_1_3_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:26_1_1_3_2"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">Def1</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>hence </b><a NAME="E4:26_1_1_3"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E7:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">Z</font> be    <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"> ( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )</span><br/><b>assume </b><a NAME="E1:26_1_1_3_3"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">Z'</font> = <font color="Maroon" title="c10">Z</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><a NAME="E3:26_1_1_3_3"/>
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Z'</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1_3"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="filter_0.html#E1:26_1_1_3_3"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:26_1_1_3_3"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">Def1</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>hence </b><a NAME="E6:26_1_1_3"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E7:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 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>
<b>then reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">F</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">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="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> &amp; ( for <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> ) )</span><br/>

<a NAME="E14:26_1_1"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:26_1_1"/>
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E7:26_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>; <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="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font></span><br/>

<b>let </b><font color="Maroon" title="c9">H</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</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">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">H</font> implies <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">H</font> )</span><br/>

<b>assume </b><a NAME="E16:26_1_1"/>
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">H</font></span><br/>

<a NAME="E17:26_1_1"/><b>then </b>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:26_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:26_1_1"/>
<font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</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>
