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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow_8.html#V1" target="_self">Baire</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:17_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E3:17_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E5:17_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:17_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="txt" href="yellow_8.html#E2:17_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_8.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:17_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E3:17_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E3:17_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T39">TOPS_3:39</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:17_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="txt" href="yellow_8.html#E1:17_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow_8.html#E6:17_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_8.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:17_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="txt" href="yellow_8.html#E9:17_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow_8.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E11:17_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="card_4.html#V1">countable</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a> ) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D3" target="_self">YELLOW_8:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:17"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>and </b><br/><a NAME="E4:17"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:17"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:17"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E3:17"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_8.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:17"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E10:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:17_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E3:17_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E4:17"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:17_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T40">TOPS_3:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:17"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E2:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow_8.html#E7:17"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E12:17"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E13:17"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T18">TOPS_3:18</a></i>;<br/>


</div>
