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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/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:73"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">F</font> is <a href="tdlat_2.html#V1">domains-family</a>
 ;<br/>

<a NAME="E2:73"/><b>then </b>
<font color="Maroon">F</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="tdlat_2.html#T65">TDLAT_2:65</a></i>;<br/>
<a NAME="E3:73"/><b>then </b>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E4:73"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="tdlat_2.html#V3">open-domains-family</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T79">TDLAT_2:79</a></i>;<br/>
<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">Y</font></span>)</span>;<br/>

<a NAME="E5:73"/><i><font color="Green">E45</font></i>: 
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="tdlat_1.html#K12">Open_Domains_Lattice</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:73"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<b>hence </b><a NAME="E7:73"/>
( <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">Y</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="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, , <a class="ref" href="tdlat_2.html#T110">TDLAT_2:110</a></i>;<br/>

<b>assume </b><a NAME="E8:73"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E9:73"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D3" target="_self">Def3</a>, <a class="ref" href="tdlat_3.html#T18" target="_self">Th18</a>, <a class="ref" href="tdlat_2.html#T93">TDLAT_2:93</a></i>;<br/>


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