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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="topgen_4.html#V4" target="_self">closed_for_countable_meets</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V1">compl-closed</a> )
 ;<br/><a NAME="E2:32_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="card_4.html#V1">countable</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be  <a href="card_4.html#V1">countable</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="txt" href="topgen_4.html#E1:32_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a>, <a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self">Def5</a></i>;<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">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:32_1_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:32_1_1_1_2"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:32_1_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topgen_4.html#E3:32_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self">Def5</a></i>;<br/><span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T11">TOPS_2:11</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font>

;<br/><b>hence </b><a NAME="E7:32_1_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E5:32_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<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">c<sub>2</sub></font> is <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self">Def4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="topgen_4.html#V3" target="_self">closed_for_countable_unions</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V1">compl-closed</a> )
 ;<br/>

<a NAME="E3:32"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="card_4.html#V1">countable</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="setfam_1.html#K6">meet</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be  <a href="card_4.html#V1">countable</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a>, <a class="txt" href="topgen_4.html#E2:32"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self">Def4</a></i>;<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">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:32_2_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:32_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topgen_4.html#E2:32"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:32_2_1_2"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:32_2_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topgen_4.html#E3:32_2_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topgen_4.html#D4" target="_self">Def4</a></i>;<br/><span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="tops_2.html#T12">TOPS_2:12</a></i>
<br/>.= 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font>

;<br/><b>hence </b><a NAME="E7:32_2_1_2"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E5:32_2_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topgen_4.html#E2:32"><i><font color="Green">E17</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<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">c<sub>2</sub></font> is <a href="topgen_4.html#V4" target="_self">closed_for_countable_meets</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:32"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topgen_4.html#D5" target="_self">Def5</a></i>;<br/>


</div>
