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

<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> &amp; <font color="Olive" title="b2">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> holds <br/><font color="Olive" title="b1">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a></span><br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</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">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> &amp; <font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> implies <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</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">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a></span><br/>

<a NAME="E3:80"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="txt" href="#E1:80"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="tops_1.html#D5" target="_self" title="TOPS_1:def.5">Def5</a></i>;<br/>
<a NAME="E4:80"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="tops_1.html#D5" target="_self" title="TOPS_1:def.5">Def5</a></i>;<br/>
<a NAME="E5:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font></span>)</span> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="txt" href="#E3:80"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="tops_1.html#T85" target="_self" title="TOPS_1:th.85">Th85</a></i>;<br/>
<a NAME="E6:80"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font></span>)</span> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></i>;<br/>
<b>hence </b><a NAME="E7:80"/>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D5" target="_self" title="TOPS_1: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>
