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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/>



<a NAME="E1:76"/>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff <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#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E2:76"/><b>then </b>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff  <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> <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="E3:76"/><b>then </b>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:76"/><b>then </b>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> )
 
;<br/>
<a NAME="E5:76"/><b>then </b>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:76"/>
( <font color="Maroon">R</font> is <a href="tops_1.html#V2" target="_self">boundary</a> iff  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>


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