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<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/> <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a></span><br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> )</span><br/>



<b>assume </b><a NAME="E1:90"/>
<font color="Maroon" title="c2">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a></span><br/>

<a NAME="E2:90"/><b>then </b>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
;<br/>
<a NAME="E3:90"/><b>then </b>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 
;<br/>
<b>hence </b><a NAME="E4:90"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">P</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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