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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</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">T</font>;<br/>



<b>assume </b><a NAME="E1:107"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F</font> is <a href="tdlat_2.html#V2" target="_self">closed-domains-family</a>
 ;<br/>

<a NAME="E2:107"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">F</font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:107"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:107"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T29">TOPS_2:29</a></i>;<br/>
<a NAME="E5:107"/>
 <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> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:107"/><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> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span></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="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E7:107"/><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> <span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E8:107"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span></span>)</span></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="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>let </b><font color="Maroon">X</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">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E9:107"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<b>hence </b><a NAME="E10:107"/>
( <font color="Maroon">X</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">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">T</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="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E11:107"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 <b>by </b><i>, , </i>;<br/>


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