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

<b>let </b><font color="Maroon" title="c1">GX</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/>( <font color="Olive" title="b1">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> iff <font color="Olive" title="b1">R</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies ( <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> iff <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:99"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c2">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:99"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c2">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> iff <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )</span><br/>

<b>thus </b><a NAME="E3:99"/>
( <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> iff <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<a NAME="E1:99_1"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="#E2:99"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E2:99_1"/>
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="tops_1.html#T55" target="_self" title="TOPS_1:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E3:99_1"/>
( <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> iff <font color="Maroon" title="c2">R</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 <b>by </b><i><a class="txt" href="#E1:99_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a>, <a class="ref" href="tops_1.html#D8" target="_self" title="TOPS_1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
