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<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="E1:54"/>
( <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> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E29</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>thus </b><a NAME="E2:54_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 ;<br/>

<a NAME="E2:54_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E1:54_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tdlat_3.html#T35" target="_self">Th35</a></i>;<br/>


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

<b>thus </b><a NAME="E3:54_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 ;<br/>

<a NAME="E2:54_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T106">TOPS_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E1:54_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tdlat_3.html#T35" target="_self">Th35</a></i>;<br/>


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


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

<b>assume </b><a NAME="E2:54"/><i><font color="Green">E29</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> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> ) )
 ;<br/>

<a NAME="E3:54"/>
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> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </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="E1:54_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/>

<a NAME="E2:54_2"/><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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <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">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<a NAME="E3:54_2"/><b>then </b><i><font color="Green">E30</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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:54_2"/>
(  <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> is <a href="tops_1.html#V6">open_condensed</a> &amp;  <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> is <a href="tops_1.html#V5">closed_condensed</a> )
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T22">TDLAT_1:22</a>, <a class="ref" href="tdlat_1.html#T23">TDLAT_1:23</a></i>;<br/>
<a NAME="E5:54_2"/><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> is <a href="tops_1.html#V5">closed_condensed</a> &amp;  <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> is <a href="tops_1.html#V6">open_condensed</a> )
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E2:54"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E6:54_2"/><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> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <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> <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="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a>, <a class="ref" href="tops_1.html#D8">TOPS_1:def 8</a></i>;<br/>
<a NAME="E7:54_2"/><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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <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> <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">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<a NAME="E8:54_2"/><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">c<sub>2</sub></font></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> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T1">TDLAT_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:54_2"/>
 <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>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E3:54_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
<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#T36" target="_self">Th36</a></i>;<br/>


</div>
