<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>
<a NAME="E1:44"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #)<br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E2:44"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #)<br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <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"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:44"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V10">Lattice-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a>,<a class="ref" href="lattices.html#D10">LATTICES:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <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" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font>, <font color="Maroon">C</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:44_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a>, <a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a>, </i>
;<br/>


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

<b>thus </b><a NAME="E3:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <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" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E3:44_1_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:44_1_3"/><b>then </b>
 ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 
;<br/>
<a NAME="E5:44_1_3"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <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> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>;<br/>
<a NAME="E6:44_1_3"/>
 <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T46">TOPS_1:46</a></i>;<br/>
<a NAME="E7:44_1_3"/><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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> &amp; <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> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T51">PRE_TOPC:51</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:44_1_3"/><b>then </b><i><font color="Green">E30</font></i>: 
 <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a>, <a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>


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

<b>thus </b><a NAME="E4:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D6">LATTICES:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <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" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> = 
 <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"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>
;<br/>


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

<b>thus </b><a NAME="E5:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font>, <font color="Maroon">C</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_1_5"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">E</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">F</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  )
 
;<br/>
<a NAME="E3:44_1_5"/><b>then </b><i><font color="Green">E41</font></i>: 
(  ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> ) &amp;  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Olive">E</font> is <a href="tops_1.html#V5">closed_condensed</a> ) &amp;  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Olive">F</font> is <a href="tops_1.html#V5">closed_condensed</a> ) )
 
;<br/>
<a NAME="E4:44_1_5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font> <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"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:44_1_5"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font></span>)</span> = 
 <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p5">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tdlat_1.html#T17" target="_self">Th17</a></i>
;<br/>


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

<b>thus </b><a NAME="E6:44_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <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" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_1_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:44_1_6"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:44_1_6"/><b>then </b><i><font color="Green">E74</font></i>: 
 ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 
;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
 <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a>, <a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) as   <a href="lattices.html#NM1">Lattice</a> ;<br/>
<a NAME="E5:44"/>
<font color="Maroon">L</font> is <a href="lattices.html#V17">Boolean</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_2"/>
<font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a>,<a class="ref" href="lattices.html#D20">LATTICES:def 20</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44_2_1"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">c</font>, <font color="Maroon">A</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<b>thus </b><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
 <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"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>
;<br/>

<b>hence </b><a NAME="E6:44_2_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:44_2"/>
<font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:44_2_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44_2_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">c</font>, <font color="Maroon">A</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<b>thus </b><font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> = 
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>

<b>hence </b><a NAME="E6:44_2_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:44_2"/>
<font color="Maroon">L</font> is <a href="lattices.html#V16">complemented</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_3"><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 <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</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" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_2_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:44_2_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">D</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 ;<br/>
<a NAME="E5:44_2_3"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">D</font> is <a href="tops_1.html#V4">condensed</a> &amp; <font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tops_1.html#T106">TOPS_1:106</a></i>;<br/>
<a NAME="E6:44_2_3"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, <a class="txt" href="tdlat_1.html#E2:44"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E7:44_2_3"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">K</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<a NAME="E9:44_2_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E10:44_2_3"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">K</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E12:44_2_3"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_2_3_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">K</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:44_2_3_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 ex <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 
;<br/>
<b>thus </b>the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">v</font> = 
<font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">v</font>

<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a>, <a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="tops_1.html#T27">TOPS_1:27</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T17" target="_self">Th17</a>, <a class="ref" href="lattice2.html#T25">LATTICE2:25</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a><br/>

<b>hence </b><a NAME="E14:44_2_3"/>
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E15:44_2_3"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E16:44_2_3"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">K</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E18:44_2_3"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<b>thus </b>the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">v</font> = 
<font color="Maroon">c</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">v</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">v</font>

;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> = 
 <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"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="lattice2.html#T22">LATTICE2:22</a></i>
;<br/>

<b>hence </b><a NAME="E20:44_2_3"/>
<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:44_2"/>
<font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font>, <font color="Maroon">C</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font> ;<br/>
<a NAME="E2:44_2_4"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">D</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">E</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">F</font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:44_2_4"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">D</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 ;<br/>
<b>consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:44_2_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">E</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 <b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:44_2_4"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">F</font> is <a href="tops_1.html#V5">closed_condensed</a> )
 <b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E9:44_2_4"/>
( <font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, <a class="ref" href="tdlat_1.html#T17" target="_self">Th17</a>, , <a class="ref" href="tops_1.html#T106">TOPS_1:106</a></i>;<br/>
<a NAME="E10:44_2_4"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E11:44_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E12:44_2_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E13:44_2_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
 <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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
 <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"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tdlat_1.html#T17" target="_self">Th17</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a>, <a class="ref" href="tdlat_1.html#T18" target="_self">Th18</a>, <a class="ref" href="tdlat_1.html#T19" target="_self">Th19</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:44"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" target="_self">Closed_Domains_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" target="_self">CLD-Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" target="_self">CLD-Meet</a> <font color="Maroon">T</font></span>)</span> #) is   <a href="lattices.html#NM8">B_Lattice</a>
 ;<br/>


</div>
