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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a></span><br/>

<span class="kw">set </span><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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #);<br/>
<a NAME="E1:9"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q9</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> = 
<font color="Maroon" title="c6">q9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c5">p9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:9"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c7">q9</font>, <font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">k9</font>, <font color="Maroon" title="c10">l9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_2"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_2"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q9</font>
 <span class="kw">and </span><br/><a NAME="E4:9_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r9</font>
 <span class="kw">and </span><br/><a NAME="E5:9_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k9</font>
 <span class="kw">and </span><br/><a NAME="E6:9_2"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l9</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c9">k9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E5:9_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E4:9_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E5:9_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c7">q9</font></span>)</span> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<font color="Maroon" title="c10">l9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E3:9_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E6:9_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E6:9_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:9"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font>, <font color="Maroon" title="c7">k9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_3"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_3"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q9</font>
 <span class="kw">and </span><br/><a NAME="E4:9_3"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k9</font>
 ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> = 
<font color="Maroon" title="c7">k9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E4:9_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c6">q9</font></span>)</span> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_3"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E4:9_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c4">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="xboole_1.html#T22" title="XBOOLE_1:th.22">XBOOLE_1:22</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:9"/><span class="lab"><font color="Green" title="E11">A14</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_4"/><span class="lab"><font color="Green" title="E12">A15</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_4"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q9</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font> = 
<font color="Maroon" title="c6">q9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c5">p9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_4"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="txt" href="#E3:9_4"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_4"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="txt" href="#E3:9_4"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E12">A17</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">p9</font>, <font color="Maroon" title="c7">q9</font>, <font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">k9</font>, <font color="Maroon" title="c10">l9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_5"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_5"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q9</font>
 <span class="kw">and </span><br/><a NAME="E4:9_5"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r9</font>
 <span class="kw">and </span><br/><a NAME="E5:9_5"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k9</font>
 <span class="kw">and </span><br/><a NAME="E6:9_5"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l9</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c9">k9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_5"><span class="lab"><font color="Green" title="E13">A18</font></span></a>, <a class="txt" href="#E5:9_5"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_5"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="txt" href="#E4:9_5"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E5:9_5"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c7">q9</font></span>)</span> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c10">l9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_5"><span class="lab"><font color="Green" title="E13">A18</font></span></a>, <a class="txt" href="#E3:9_5"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="txt" href="#E6:9_5"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_5"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E6:9_5"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:9"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font>, <font color="Maroon" title="c7">k9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:9_6"/><span class="lab"><font color="Green" title="E13">A23</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E3:9_6"/><span class="lab"><font color="Green" title="E14">A24</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q9</font>
 <span class="kw">and </span><br/><a NAME="E4:9_6"/><span class="lab"><font color="Green" title="E15">A25</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k9</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">q</font></span>)</span> = 
<font color="Maroon" title="c5">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c7">k9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_6"><span class="lab"><font color="Green" title="E13">A23</font></span></a>, <a class="txt" href="#E4:9_6"><span class="lab"><font color="Green" title="E15">A25</font></span></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c5">p9</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p9</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_6"><span class="lab"><font color="Green" title="E13">A23</font></span></a>, <a class="txt" href="#E3:9_6"><span class="lab"><font color="Green" title="E14">A24</font></span></a>, <a class="txt" href="#E4:9_6"><span class="lab"><font color="Green" title="E15">A25</font></span></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_6"><span class="lab"><font color="Green" title="E13">A23</font></span></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:9"/><span class="kw">then </span>
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_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="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E11">A14</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" title="LOPCLSET:func.4">T_join</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" title="LOPCLSET:func.5">T_meet</a> <font color="Maroon" title="c1">T</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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