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<div class="add">

<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="E1:105"/><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/>

<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#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:105"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E3:105"/>
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_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="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">A</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>;<br/>
<a NAME="E1:105_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i>, <a class="ref" href="tdlat_2.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E2:105_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">C</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:105_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E5:105_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T26" target="_self">Th26</a></i>;<br/>
<b>assume </b><a NAME="E2:105_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:105_1_1"/><b>then </b>
<font color="Maroon">B</font> <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#K5">union</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:105_1_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="tdlat_2.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:105_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span>  st <font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">a</font> <a href="lattices.html#R1">[=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E2:105_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E3:105_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">C</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E4:105_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Olive">C</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 
;<br/>
<b>assume </b><a NAME="E5:105_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E6:105_1_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>assume </b><a NAME="E2:105_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:105_1_2_1"/><b>then </b>
<font color="Maroon">C1</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i>, <a class="ref" href="tdlat_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:105_1_2_1"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">P</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E5:105_1_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E7:105_1_2_1"/>
<font color="Maroon">c</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E8:105_1_2_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="tdlat_2.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:105_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:105_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="tdlat_2.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:105_1"/>
 <a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">T</font> is <a href="lattice3.html#V4">complete</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:105_1"/>
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_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="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>


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


</div>
