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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:101"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <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">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<a NAME="E1:101_1"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T18" title="TDLAT_1:th.18">TDLAT_1:18</a></i>;<br/>
<a NAME="E2:101_1"/><b>then </b><i><font color="Green" title="E63">A1</font></i>: 
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:101_1"/><b>then </b>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5" title="TDLAT_1:def.5">TDLAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">e</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self" title="TDLAT_2:th.94">Th94</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">E</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="tdlat_2.html#E2:101_1"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="tdlat_1.html#D5" title="TDLAT_1:def.5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E6:101_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span> holds  <font color="Maroon" title="c2">e</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self" title="TDLAT_2:th.94">Th94</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">e</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c3">E</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="tdlat_2.html#T95" target="_self" title="TDLAT_2:th.95">Th95</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:101_1"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <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">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T21" title="LATTICE2:th.21">LATTICE2:21</a></i>;<br/>


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

<b>thus </b><a NAME="E2:101"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <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">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>proof </b></a><div class="add">

<a NAME="E1:101_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T19" title="TDLAT_1:th.19">TDLAT_1:19</a></i>;<br/>
<a NAME="E2:101_2"/><b>then </b><i><font color="Green" title="E63">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:101_2"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5" title="TDLAT_1:def.5">TDLAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">e</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self" title="TDLAT_2:th.94">Th94</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">E</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="tdlat_2.html#E2:101_2"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="tdlat_1.html#D5" title="TDLAT_1:def.5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E6:101_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span> holds  <font color="Maroon" title="c2">e</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self" title="TDLAT_2:th.94">Th94</a></i>;<br/>
<a NAME="E2:101_2_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E3:101_2_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5" title="TDLAT_1:def.5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E4:101_2_1"/><b>then </b><i><font color="Green" title="E64">A3</font></i>: 
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b1">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 
;<br/>
<b>thus </b><font color="Maroon" title="c2">e</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> = 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_2.html#T95" target="_self" title="TDLAT_2:th.95">Th95</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T14" title="PRE_TOPC:th.14">PRE_TOPC:14</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="tdlat_2.html#E4:101_2_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="tops_1.html#D7" title="TOPS_1:def.7">TOPS_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:101_2"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <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">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T24" title="LATTICE2:th.24">LATTICE2:24</a></i>;<br/>


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


</div>
