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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="E1:49"/>
( <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="pre_topc.html#V4">closed</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> <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="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E25</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="E2:49_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:49_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E4:49_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </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"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E1:49_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tdlat_3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E5:49_1"/><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> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#D1">TOPS_1:def 1</a></i>;<br/>
<a NAME="E6:49_1"/><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:49_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> <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#D1">TOPS_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E25</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="pre_topc.html#V4">closed</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> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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="E1:49_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:49_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> <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;  <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="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tdlat_3.html#E2:49"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
<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#T29" target="_self">Th29</a></i>;<br/>


</div>
