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

<b>let </b><font color="Maroon">T</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">T</font>;<br/>



<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">A</font> is <a href="isomichi.html#V1" target="_self">supercondensed</a> implies (  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> is <a href="tops_1.html#V6">regular_open</a> &amp;  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> is <a href="isomichi.html#V1" target="_self">supercondensed</a>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E4:44_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1"/>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> is <a href="tops_1.html#V6">regular_open</a> &amp;  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E31</font></i>: 
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> is <a href="tops_1.html#V6">regular_open</a> &amp;  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<a NAME="E3:44"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D8">TOPS_1:def 8</a></i>;<br/>
<a NAME="E4:44"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:44"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E6:44"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:44"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/>
<a NAME="E8:44"/>
<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="E9:44"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</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="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E10:44"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:44"/>
<font color="Maroon">A</font> is <a href="isomichi.html#V1" target="_self">supercondensed</a>
 <b>by </b><i/>;<br/>


</div>
