<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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">A</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  implies <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:26"/>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  or <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )</span><br/>

<a NAME="E2:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D5" title="TOPS_1:def.5">TOPS_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:26"/>
<font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> </span><br/>

<a NAME="E4:26"/><span class="kw">then </span>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="tops_1.html#T85" title="TOPS_1:th.85">TOPS_1:85</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:26"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T11" target="_self" title="TOPS_3:th.11">Th11</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
