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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <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">B</font>, <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b2">A</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">B</font>, <font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> )</span><br/>





<b>assume </b><a NAME="E1:23"/>
<font color="Maroon" title="c2">B</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"> (  not <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> )</span><br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green" title="E16">A1</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D5" title="TOPS_1:def.5">TOPS_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E3:23"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a></span><br/>

<a NAME="E4:23"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tops_3.html#T11" target="_self" title="TOPS_3:th.11">Th11</a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<b>hence </b><a NAME="E5:23"/>
<font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D5" title="TOPS_1:def.5">TOPS_1:def 5</a></i>; <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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