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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="E1:92"/>
<font color="Maroon">R</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:92"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</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#T52">PRE_TOPC:52</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:92_1"/>
<font color="Maroon">R</font> is <a href="tops_1.html#V3" target="_self">nowhere_dense</a>
 ;<br/><a NAME="E2:92_1"/><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="E3:92_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<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="E4:92_1"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, <a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E5:92_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:92"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E5:92"/><i><font color="Green">E37</font></i>: 
 <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:92"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T52" target="_self">Th52</a>, </i>;<br/>
<a NAME="E7:92"/><b>then </b>
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#T49" target="_self">Th49</a>, <a class="ref" href="tops_1.html#T53" target="_self">Th53</a>, <a class="ref" href="xboole_1.html#T38">XBOOLE_1:38</a></i>;<br/>
<a NAME="E8:92"/><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#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E9:92"/>
<font color="Maroon">R</font> is <a href="tops_1.html#V3" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T45" target="_self">Th45</a></i>;<br/>


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