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

<b>let </b><font color="Maroon">X</font> be   <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>thus </b><a NAME="E1:12"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> implies for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:12_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:12_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<b>assume </b><a NAME="E4:12_1"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E6:12_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="tops_1.html#T86">TOPS_1:86</a></i>;<br/>
<b>hence </b><a NAME="E7:12_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:12_2"/>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><a NAME="E2:12_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><a NAME="E3:12_2"/><b>then </b>
<font color="Maroon">G</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T86">TOPS_1:86</a></i>;<br/>


</div>
