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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:72"/>
<font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a>
 ;<br/>

<a NAME="E2:72"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T65" title="TDLAT_2:th.65">TDLAT_2:65</a></i>;<br/>
<a NAME="E3:72"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T41" target="_self" title="TDLAT_3:th.41">Th41</a></i>;<br/>
<a NAME="E4:72"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V2" title="TDLAT_2:attr.2">closed-domains-family</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T72" title="TDLAT_2:th.72">TDLAT_2:72</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c3">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8" title="TDLAT_1:func.8">Closed_Domains_Lattice</a> <font color="Maroon" title="c1">Y</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_3.html#T43" target="_self" title="TDLAT_3:th.43">Th43</a></i>;<br/>
<b>assume </b><a NAME="E6:72"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>thus </b> <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c3">S</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">Y</font></span>)</span> = 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">P</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K8" title="TDLAT_1:func.8">Closed_Domains_Lattice</a> <font color="Maroon" title="c1">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T43" target="_self" title="TDLAT_3:th.43">Th43</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_3.html#E4:72"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="tdlat_3.html#E6:72"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="tdlat_2.html#T100" title="TDLAT_2:th.100">TDLAT_2:100</a></i>
;<br/>


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