<?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="E1:14"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> implies for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> holds <br/> <a href="tops_1.html#K1">Int</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E13</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_3.html#D1" target="_self">TOPS_3:def 1</a><br/>

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

<b>assume </b><a NAME="E2:14_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:14_1"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:14_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_3.html#T3" target="_self">Th3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:14"/>
for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> holds <br/> <a href="tops_1.html#K1">Int</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E3:14"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> )
 
;<br/>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i/>;<br/>


</div>
