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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E3:64_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : <font color="Olive">D</font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/><a NAME="E4:64_1"/><b>then </b>
 ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Olive">D</font> is <a href="tops_1.html#V4">condensed</a> )
 ;<br/><a NAME="E5:64_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="tops_1.html#V5">closed_condensed</a>
 <b>by </b><i/>;<br/><a NAME="E6:64_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : <font color="Olive">E</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:64_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:64"/><b>then </b>
(  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">Y</font> &amp;  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="tdlat_1.html#T31">TDLAT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E3:64"/>
 <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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