<?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="E1:34"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V5">regular_closed</a> implies ( <font color="Maroon">A</font> is <a href="isomichi.html#V2" target="_self">subcondensed</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V5">regular_closed</a>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <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> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
( <font color="Maroon">A</font> is <a href="isomichi.html#V2" target="_self">subcondensed</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:34"/>
( <font color="Maroon">A</font> is <a href="isomichi.html#V2" target="_self">subcondensed</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E3:34"/><b>then </b>
(  <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> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E4:34"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V5">regular_closed</a>
 ;<br/>


</div>
