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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:30"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<a NAME="E2:30"/><b>then </b><i><font color="Green">E13</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#D5">TOPS_1:def 5</a></i>;<br/>
<a NAME="E3:30"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<b>assume </b><a NAME="E5:30"/>
<font color="Maroon">B</font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<a NAME="E6:30"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="tops_1.html#V2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_1.html#T85">TOPS_1:85</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_3.html#T2" target="_self">Th2</a></i>;<br/>


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