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<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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:42"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</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#D5">TOPS_1:def 5</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/>
<a NAME="E4:42_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T2">TDLAT_3:2</a></i>;<br/>
<a NAME="E5:42_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E6:42_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:42_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:42"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E3:42"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tops_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:42"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:42"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E6:42"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T2">TDLAT_3:2</a></i>;<br/>
<a NAME="E7:42"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</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#T84">TOPS_1:84</a></i>;<br/>
<b>hence </b><a NAME="E8:42"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D5">TOPS_1:def 5</a></i>;<br/>


</div>
