<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">US</font> =  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">STP</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span> ;<br/>
<a NAME="E1:46_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<a NAME="E2:46_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">B</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:46_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">STP</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E3:46_1_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) is   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">TS</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #);<br/>
<a NAME="E1:46_1_1_2"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> ;<br/>
<a NAME="E1:46_1_1_2_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:46_1_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:46_1_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:46_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T67" onmouseover="rs('zfmisc_1/T67')" onmouseout="rh()">ZFMISC_1:67</a></span>;<br/>
<a NAME="E3:46_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T68" onmouseover="rs('zfmisc_1/T68')" onmouseout="rh()">ZFMISC_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">SB</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_2_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">SB</font> = <font color="Maroon" title="c7">SB</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E6:46_1_1_2_1"/>
 <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_1_2"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_1_2"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c10">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> 
 ;<br/>
<a NAME="E5:46_1_1_2_1_2"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T11" onmouseover="rs('filter_0/T11')" onmouseout="rh()">FILTER_0:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46_1_1_2_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_1_2"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E4:46_1_1_2_1_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:46_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">SB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/>
<a NAME="E8:46_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="latticea.html#K3" title="LATTICEA:func.3">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#D5" target="_self" onmouseover="rs('latticea/D5')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E9:46_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">SB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="ref" href="latticea.html#T28" target="_self" onmouseover="rs('latticea/T28')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:46_1_1_2_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_2_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:46_1_1_2"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_2"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> </span> } </span> ;<br/>
<a NAME="E2:46_1_1_2_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> </span> } </span>  or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<a NAME="E2:46_1_1_2_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E4:46_1_1_2_2_1"/>
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:46_1_1_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">BS</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> </span> } </span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E4:46_1_1_2_2"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T54" onmouseover="rs('eqrel_1/T54')" onmouseout="rh()">EQREL_1:54</a></span>;<br/>
<a NAME="E5:46_1_1_2_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_2_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> </span><br/>

<a NAME="E2:46_1_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">STP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">C</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_2_2_2"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_2_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 ;<br/>
<a NAME="E6:46_1_1_2_2_2"/>
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_2_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E4:46_1_1_2_2_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E5:46_1_1_2_2_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:46_1_1_2_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_2_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:46_1_1_2_2"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_2_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_2_3"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_2_3"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font>
 ;<br/>
<a NAME="E5:46_1_1_2_2_3"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>( <font color="Olive" title="b1">D</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c10">C</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_2_3"><span class="lab"><font color="Green" title="E26">A15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46_1_1_2_2_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_2_3"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:46_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">BS</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a></span>;<br/>
<a NAME="E8:46_1_1_2_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">BS</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:46_1_1_2_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E7:46_1_1_2_2"><span class="lab"><font color="Green" title="E25">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:46_1_1_2"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) holds <br/><font color="Olive" title="b1">p</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) &amp; <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) implies <font color="Maroon" title="c6">p</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46_1_1_2_3"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 <span class="kw">and </span><br/><a NAME="E2:46_1_1_2_3"/><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">P</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:46_1_1_2_3"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_3"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_3"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:46_1_1_2_3"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">and </span><br/><a NAME="E8:46_1_1_2_3"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
<font color="Maroon" title="c9">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1_2_3"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/>
<a NAME="E9:46_1_1_2_3"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T28" onmouseover="rs('setfam_1/T28')" onmouseout="rh()">SETFAM_1:28</a></span>;<br/>
<a NAME="E10:46_1_1_2_3"/>
 <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>) or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">X</font>, <font color="Maroon" title="c12">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_3_1"/><span class="lab"><font color="Green" title="E29">A24</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_3_1"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<font color="Maroon" title="c12">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_3_1"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D5" onmouseover="rs('setfam_1/D5')" onmouseout="rh()">SETFAM_1:def 5</a></span>;<br/>
<a NAME="E6:46_1_1_2_3_1"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T29" onmouseover="rs('xboole_1/T29')" onmouseout="rh()">XBOOLE_1:29</a></span>;<br/>
<a NAME="E7:46_1_1_2_3_1"/>
<font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_3_1"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E4:46_1_1_2_3_1"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E8:46_1_1_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1_2_3_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:46_1_1_2_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">C</font> =  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> (<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font>) as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c10">C</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E13:46_1_1_2_3"/>
<font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_3_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">X</font>, <font color="Maroon" title="c14">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
<font color="Maroon" title="c14">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E31">A30</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D5" onmouseover="rs('setfam_1/D5')" onmouseout="rh()">SETFAM_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latticea.html#K3" title="LATTICEA:func.3">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E3:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="latticea.html#T28" target="_self" onmouseover="rs('latticea/T28')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latticea.html#K3" title="LATTICEA:func.3">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46_1_1_2_3"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E4:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="latticea.html#T28" target="_self" onmouseover="rs('latticea/T28')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E10:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="latticea.html#D5" target="_self" onmouseover="rs('latticea/D5')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E11:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
<font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="latticea.html#D5" target="_self" onmouseover="rs('latticea/D5')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E12:46_1_1_2_3_2"/><span class="lab"><font color="Green" title="E36">A35</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:46_1_1_2_3_2_1"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> </span><br/>

<a NAME="E2:46_1_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:46_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">F1</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:46_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F1</font>
 <span class="kw">and </span><br/><a NAME="E6:46_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E42">A41</font></span>: 
<font color="Maroon" title="c18">F1</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> 
 <span class="kw">and </span><br/><a NAME="E7:46_1_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E43">A42</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E2:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a></span>;<br/>
<a NAME="E8:46_1_1_2_3_2_1_1"/>
 ex <font color="Olive" title="b1">F2</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F2</font> &amp; <font color="Olive" title="b1">F2</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E3:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E40">A39</font></span></a></span>;<br/>
<a NAME="E9:46_1_1_2_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E7:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E43">A42</font></span></a>, <a class="ref" href="filter_0.html#T8" onmouseover="rs('filter_0/T8')" onmouseout="rh()">FILTER_0:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:46_1_1_2_3_2_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E6:46_1_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E42">A41</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:46_1_1_2_3_2_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">F0</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E38">A43</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F0</font>
 <span class="kw">and </span><br/><a NAME="E4:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E39">A44</font></span>: 
<font color="Maroon" title="c18">F0</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> 
 <span class="kw">and </span><br/><a NAME="E5:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E40">A45</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">F0</font>
 ;<br/>
<a NAME="E6:46_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">F0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="ref" href="filter_0.html#T8" onmouseover="rs('filter_0/T8')" onmouseout="rh()">FILTER_0:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E8:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E41">A46</font></span>: 
<font color="Maroon" title="c19">F</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c18">F0</font>
 <span class="kw">and </span><br/><a NAME="E9:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E42">A47</font></span>: 
<font color="Maroon" title="c19">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> 
 <span class="kw">and </span><br/><a NAME="E10:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E43">A48</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E39">A44</font></span></a></span>;<br/>
<a NAME="E11:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E44">A49</font></span>: 
<font color="Maroon" title="c19">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E9:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E42">A47</font></span></a>, <a class="txt" href="#E10:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E43">A48</font></span></a></span>;<br/>
<a NAME="E12:46_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">F0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="ref" href="filter_0.html#T8" onmouseover="rs('filter_0/T8')" onmouseout="rh()">FILTER_0:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">F1</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E14:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E45">A50</font></span>: 
<font color="Maroon" title="c20">F1</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c18">F0</font>
 <span class="kw">and </span><br/><a NAME="E15:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E46">A51</font></span>: 
<font color="Maroon" title="c20">F1</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> 
 <span class="kw">and </span><br/><a NAME="E16:46_1_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E47">A52</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E39">A44</font></span></a></span>;<br/>
<a NAME="E17:46_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c20">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E15:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E46">A51</font></span></a>, <a class="txt" href="#E16:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E47">A52</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:46_1_1_2_3_2_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E38">A43</font></span></a>, <a class="txt" href="#E8:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="txt" href="#E11:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E44">A49</font></span></a>, <a class="txt" href="#E14:46_1_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E45">A50</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:46_1_1_2_3_2_1"/>
<font color="Maroon" title="c13">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2_3_2_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:46_1_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="latticea.html#K3" title="LATTICEA:func.3">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c15">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c16">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#D5" target="_self" onmouseover="rs('latticea/D5')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:46_1_1_2_3_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E12:46_1_1_2_3_2"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="latticea.html#T28" target="_self" onmouseover="rs('latticea/T28')" onmouseout="rh()">Th23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:46_1_1_2_3"/>
<font color="Maroon" title="c6">p</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1_2_3"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E7:46_1_1_2_3"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E9:46_1_1_2_3"><span class="lab"><font color="Green" title="E28">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:46_1_1_2"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) is   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:46_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">TS</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">STP</font> #) as   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">TS</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">TS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> &amp;  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">TS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:46_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">TS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font> &amp;  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">TS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="latticea.html#K1" title="LATTICEA:func.1">PFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="latticea.html#K4" title="LATTICEA:func.4">StoneR</a> <font color="Maroon" title="c1">L</font> </span> } </span>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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