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

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a></span><br/>

<b>set </b><font color="Maroon" title="c2">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #);<br/>
<a NAME="E1:42"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>,<a class="ref" href="lattices.html#D10" title="LATTICES:def.10">LATTICES:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">b</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> ;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> = 
<font color="Maroon" title="c6">B</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_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c5">c</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> ;<br/>
<a NAME="E2:42_1_2"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>;<br/>
<a NAME="E3:42_1_2"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:42_1_2"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="#E3:42_1_2"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">b</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> ;<br/>
<a NAME="E2:42_1_3"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>;<br/>
<a NAME="E3:42_1_3"/>
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:42_1_3"/><b>then </b>
 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="c6">B</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/>
<a NAME="E5:42_1_3"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D7" title="TOPS_1:def.7">TOPS_1:def 7</a></i>;<br/>
<a NAME="E6:42_1_3"/>
 <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="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T46" title="TOPS_1:th.46">TOPS_1:46</a></i>;<br/>
<a NAME="E7:42_1_3"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
(  <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="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="#E5:42_1_3"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="pre_topc.html#T51" title="PRE_TOPC:th.51">PRE_TOPC:51</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="#E2:42_1_3"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="txt" href="#E7:42_1_3"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E4:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">b</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> ;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</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="c6">B</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_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E5:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c5">c</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> ;<br/>
<a NAME="E2:42_1_5"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</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">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</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">F</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  )
 
;<br/>
<a NAME="E3:42_1_5"/><b>then </b><i><font color="Green" title="E34">A6</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="c6">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> ) &amp;  ex <font color="Olive" title="b1">E</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">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp;  ex <font color="Olive" title="b1">F</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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) )
 
;<br/>
<a NAME="E4:42_1_5"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>;<br/>
<a NAME="E5:42_1_5"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
 <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="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:42_1_5"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:42_1_5"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="tdlat_1.html#T28" target="_self" title="TDLAT_1:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="#E5:42_1_5"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E6:42_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">b</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> ;<br/>
<a NAME="E2:42_1_6"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>;<br/>
<a NAME="E3:42_1_6"/>
<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">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:42_1_6"/><b>then </b><i><font color="Green" title="E35">A10</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="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> = 
 <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="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:42_1_6"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</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="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="#E4:42_1_6"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="tops_1.html#D7" title="TOPS_1:def.7">TOPS_1:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<a NAME="E3:42"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_2"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a>,<a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_2_1"/><i><font color="Green" title="E34">A11</font></i>: 
 <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" target="_self" title="TDLAT_1:th.18">Th18</a></i>;<br/>
<a NAME="E2:42_2_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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">c</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 <font color="Maroon" title="c3">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> holds <br/> ( <font color="Maroon" title="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">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> ;<br/>
<b>thus </b><font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">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="c6">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="#E1:42_2_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="tops_1.html#D7" title="TOPS_1:def.7">TOPS_1:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font></span><br/>

<b>hence </b><a NAME="E6:42_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:42_2"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:42_2_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" target="_self" title="TDLAT_1:th.19">Th19</a></i>;<br/>
<a NAME="E2:42_2_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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">c</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 <font color="Maroon" title="c3">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> holds <br/> ( <font color="Maroon" title="c4">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">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> ;<br/>
<b>thus </b><font color="Maroon" title="c4">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c6">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font></span><br/>

<b>hence </b><a NAME="E6:42_2_2"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:42_2"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">b</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> ;<br/>
<a NAME="E2:42_2_3"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:42_2_3"/><i><font color="Green" title="E34">A12</font></i>: 
( <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 ;<br/>
<a NAME="E5:42_2_3"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c6">D</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> &amp; <font color="Maroon" title="c6">D</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="#E4:42_2_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="tops_1.html#T106" title="TOPS_1:th.106">TOPS_1:106</a></i>;<br/>
<a NAME="E6:42_2_3"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 
<b>by </b><i><a class="txt" href="#E4:42_2_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="tdlat_1.html#T16" target="_self" title="TDLAT_1:th.16">Th16</a>, <a class="ref" href="tdlat_1.html#T24" target="_self" title="TDLAT_1:th.24">Th24</a></i>;<br/>
<a NAME="E7:42_2_3"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K</font> where <font color="Olive" title="b1">K</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">K</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E9:42_2_3"/>
 <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" target="_self" title="TDLAT_1:th.19">Th19</a></i>;<br/>
<a NAME="E10:42_2_3"/><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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K</font> where <font color="Olive" title="b1">K</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">K</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">c</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 <font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E12:42_2_3"/><i><font color="Green" title="E36">A14</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> holds  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c8">c</font>,<font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">v</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> ;<br/>
<a NAME="E2:42_2_3_1"/>
<font color="Maroon" title="c10">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K</font> where <font color="Olive" title="b1">K</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">K</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:42_2_3_1"/><b>then </b><i><font color="Green" title="E37">A15</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="c10">V</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>the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">v</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"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">V</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</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="c10">V</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c9">v</font>
<b>by </b><i><a class="txt" href="#E3:42_2_3_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="tops_1.html#D7" title="TOPS_1:def.7">TOPS_1:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon" title="c7">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">D</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:42_2_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="#E5:42_2_3"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:42_2_3"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>
<br/>.= 
<font color="Maroon" title="c8">c</font>
<b>by </b><i><a class="ref" href="tops_1.html#T27" title="TOPS_1:th.27">TOPS_1:27</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="#E12:42_2_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="lattice2.html#T25" title="LATTICE2:th.25">LATTICE2:25</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font> )</span><br/>

<b>hence </b><a NAME="E14:42_2_3"/>
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font> )</span><br/>

<a NAME="E15:42_2_3"/>
 <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" target="_self" title="TDLAT_1:th.18">Th18</a></i>;<br/>
<a NAME="E16:42_2_3"/><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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K</font> where <font color="Olive" title="b1">K</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">K</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">c</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 <font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E18:42_2_3"/><i><font color="Green" title="E37">A16</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> holds  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c9">c</font>,<font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">V</font> = <font color="Maroon" title="c10">v</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> ;<br/>
<b>thus </b>the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">v</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">V</font>
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c10">v</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</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="c5">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T8" target="_self" title="TDLAT_1:th.8">Th8</a></i>
<br/>.= 
<font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="#E18:42_2_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="lattice2.html#T22" title="LATTICE2:th.22">LATTICE2:22</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font></span><br/>

<b>hence </b><a NAME="E20:42_2_3"/>
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E4:42_2"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c6">c</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> ;<br/>
<a NAME="E2:42_2_4"/><i><font color="Green" title="E34">A17</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</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">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</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">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</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">F</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:42_2_4"/><i><font color="Green" title="E35">A18</font></i>: 
( <font color="Maroon" title="c10">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c10">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:42_2_4"/><i><font color="Green" title="E36">A19</font></i>: 
( <font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c11">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 <b>by </b><i><a class="txt" href="#E2:42_2_4"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:42_2_4"/><i><font color="Green" title="E37">A20</font></i>: 
( <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c12">F</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 <b>by </b><i><a class="txt" href="#E2:42_2_4"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/>
<a NAME="E9:42_2_4"/>
( <font color="Maroon" title="c10">D</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c11">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c12">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="#E4:42_2_4"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="#E6:42_2_4"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="#E8:42_2_4"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="tops_1.html#T106" title="TOPS_1:th.106">TOPS_1:106</a></i>;<br/>
<a NAME="E10:42_2_4"/><b>then </b><i><font color="Green" title="E38">A21</font></i>: 
( <font color="Maroon" title="c10">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c10">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>
<a NAME="E11:42_2_4"/><i><font color="Green" title="E39">A22</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>;<br/>
<a NAME="E12:42_2_4"/><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>;<br/>
<a NAME="E13:42_2_4"/><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
 <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="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E11:42_2_4"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="tdlat_1.html#D7" target="_self" title="TDLAT_1:def.7">Def7</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:42_2_4"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="#E6:42_2_4"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="#E10:42_2_4"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="tdlat_1.html#T6" target="_self" title="TDLAT_1:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:42_2_4"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="#E13:42_2_4"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="tdlat_1.html#D6" target="_self" title="TDLAT_1:def.6">Def6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK9" title="TDLAT_1:NK.9">CLD-Union</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK11" title="TDLAT_1:NK.11">CLD-Meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
