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

<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> iff ( <font color="Olive" title="b1">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> iff ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> implies ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )</span><br/><a NAME="E2:32_1"/>
for <font color="Olive" title="b1">G</font> being <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G</font> be  <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self" title="TOPGEN_4:def.5">Def5</a>, <a class="ref" href="setfam_1.html#T2" title="SETFAM_1:th.2">SETFAM_1:2</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">G'</font> = <font color="Maroon" title="c3">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:32_1_1_1_2"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="topgen_4.html#T16" target="_self" title="TOPGEN_4:th.16">Th16</a></i>;<br/><a NAME="E4:32_1_1_1_2"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self" title="TOPGEN_4:th.1">Th1</a></i>;<br/><a NAME="E5:32_1_1_1_2"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="topgen_4.html#E3:32_1_1_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self" title="TOPGEN_4:def.5">Def5</a></i>;<br/><span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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="Maroon" title="c4">G'</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T11" title="TOPS_2:th.11">TOPS_2:11</a></i>
<br/>.= 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">G'</font>

;<br/><b>hence </b><a NAME="E7:32_1_1_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="topgen_4.html#E5:32_1_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:32_1"/>
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )</span><br/>

<a NAME="E3:32"/>
for <font color="Olive" title="b1">G</font> being <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G</font> be  <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> implies  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1"/>
( <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</a>, <a class="ref" href="setfam_1.html#T2" title="SETFAM_1:th.2">SETFAM_1:2</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_2"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">G'</font> = <font color="Maroon" title="c3">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:32_2_1_2"/><i><font color="Green" title="E19">A7</font></i>: 
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="topgen_4.html#T16" target="_self" title="TOPGEN_4:th.16">Th16</a></i>;<br/><a NAME="E4:32_2_1_2"/>
 <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self" title="TOPGEN_4:th.1">Th1</a></i>;<br/><a NAME="E5:32_2_1_2"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="topgen_4.html#E3:32_2_1_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</a></i>;<br/><span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c4">G'</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T12" title="TOPS_2:th.12">TOPS_2:12</a></i>
<br/>.= 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c4">G'</font>

;<br/><b>hence </b><a NAME="E7:32_2_1_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="topgen_4.html#E5:32_2_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V4" title="TOPGEN_4:attr.4">closed_for_countable_meets</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self" title="TOPGEN_4:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
