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<b>let </b><font color="Maroon">GX</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:86"/>
<font color="Maroon">R</font> is <a href="tops_1.html#V3" target="_self">nowhere_dense</a>
 ;<br/>

<a NAME="E2:86"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E3:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1" target="_self">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E4:86"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E5:86"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<a NAME="E7:86"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T47" target="_self">Th47</a>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E8:86"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:86"/>
<font color="Maroon">R</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1" target="_self">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/>


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