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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">Baire</a>  iff for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  ) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  ) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  ) implies <font color="Maroon" title="c1">T</font> is  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">Baire</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">Baire</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  ) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  ) implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">and </span><br/><a NAME="E3:28_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c3">G</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E5:28_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c4">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/>

<a NAME="E2:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></span>;<br/>
<a NAME="E3:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:28_1_1"/>
<font color="Maroon" title="c4">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T39" title="TOPS_3:th.39">TOPS_3:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:28_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="yellow_8.html#T4" target="_self" title="YELLOW_8:th.4">Th4</a>, <a class="ref" href="yellow_8.html#T5" target="_self" title="YELLOW_8:th.5">Th5</a></span>;<br/><a NAME="E7:28_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Olive" title="b1">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E5:28_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="yellow_8.html#D3" target="_self" title="YELLOW_8:def.3">Def3</a></span>;<br/><a NAME="E8:28_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_8.html#T15" target="_self" title="YELLOW_8:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E9:28_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D4" title="TOPS_1:def.4">TOPS_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:28"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  ) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">Baire</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_8.html#D3" target="_self" title="YELLOW_8:def.3">YELLOW_8:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  &amp; ( for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) implies  ex <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">I</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:28"/>
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">G</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/><font color="Olive" title="b1">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> implies <font color="Maroon" title="c5">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_2"/>
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> </span><br/>

<a NAME="E2:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></span>;<br/>
<a NAME="E3:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:28_2"/>
<font color="Maroon" title="c5">S</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T40" title="TOPS_3:th.40">TOPS_3:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:28"/>
<font color="Maroon" title="c4">G</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="yellow_8.html#T4" target="_self" title="YELLOW_8:th.4">Th4</a>, <a class="ref" href="yellow_8.html#T5" target="_self" title="YELLOW_8:th.5">Th5</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">G</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/>
<a NAME="E11:28"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_8.html#T16" target="_self" title="YELLOW_8:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:28"/>
<font color="Maroon" title="c3">I</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T18" title="TOPS_3:th.18">TOPS_3:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
