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<span class="kw">let </span><font color="Maroon" title="c1">GX</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> holds <br/> ( <font color="Olive" title="b1">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<a NAME="E1:64"/>
( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff <font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D4" target="_self" title="TOPS_1:def.4">Def4</a></span>;<br/>
<a NAME="E2:64"/><span class="kw">then </span>
( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D3" target="_self" title="TOPS_1:def.3">Def3</a></span>;<br/>
<a NAME="E3:64"/><span class="kw">then </span>
( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T42" title="SUBSET_1:th.42">SUBSET_1:42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64"/>
( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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