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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> holds <br/> <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:53_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/>

<a NAME="E3:53_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E1:53_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tdlat_3.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E4:53_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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E1:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tdlat_3.html#E2:53_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tdlat_3.html#T36" target="_self">Th36</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> holds <br/> <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/><a NAME="E2:53_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E2:53"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:53_2"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:53"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T34" target="_self">Th34</a></i>;<br/>


</div>
