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

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

<b>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="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="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
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="10_1"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</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><b> such that </b><br/><a NAME="E2:10_1"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font> )
 ;<br/>
<b>thus </b><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">q'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c5">p'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<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>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/><i><font color="Green" title="E11">A3</font></i>: 
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="10_2"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">q'</font>, <font color="Maroon" title="c8">r'</font>, <font color="Maroon" title="c9">k'</font>, <font color="Maroon" title="c10">l'</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><b> such that </b><br/><a NAME="E2:10_2"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q'</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r'</font> &amp; <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">k'</font> &amp; <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">l'</font> )
 ;<br/>
<b>thus </b><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">p'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c9">k'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">p'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r'</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c7">q'</font></span>)</span> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c10">l'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c8">r'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<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>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green" title="E12">A5</font></i>: 
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="10_3"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</font>, <font color="Maroon" title="c7">k'</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><b> such that </b><br/><a NAME="E2:10_3"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font> &amp; <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">k'</font> )
 ;<br/>
<b>thus </b><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">k'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_3"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c6">q'</font></span>)</span> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_3"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_3"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="xboole_1.html#T22" title="XBOOLE_1:th.22">XBOOLE_1:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:10"/><i><font color="Green" title="E13">A7</font></i>: 
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="10_4"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</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><b> such that </b><br/><a NAME="E2:10_4"/><i><font color="Green" title="E14">A8</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font> )
 ;<br/>
<b>thus </b><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">q'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c5">p'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_4"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<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>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_4"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:10"/><i><font color="Green" title="E14">A9</font></i>: 
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="10_5"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">q'</font>, <font color="Maroon" title="c8">r'</font>, <font color="Maroon" title="c9">k'</font>, <font color="Maroon" title="c10">l'</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><b> such that </b><br/><a NAME="E2:10_5"/><i><font color="Green" title="E15">A10</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q'</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r'</font> &amp; <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">k'</font> &amp; <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">l'</font> )
 ;<br/>
<b>thus </b><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">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c9">k'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c6">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r'</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c7">q'</font></span>)</span> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c10">l'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c8">r'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<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>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:10"/>
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="10_6"><b>proof </b></a><div class="add">

<b>let </b><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> #);<br/>

<b>consider </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</font>, <font color="Maroon" title="c7">k'</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><b> such that </b><br/><a NAME="E2:10_6"/><i><font color="Green" title="E15">A11</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font> &amp; <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">k'</font> )
 ;<br/>
<b>thus </b><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">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <font color="Maroon" title="c7">k'</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_6"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="lopclset.html#D3" target="_self" title="LOPCLSET:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c5">p'</font> <a href="lopclset.html#K3" title="LOPCLSET:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p'</font> <a href="lopclset.html#K2" title="LOPCLSET:func.2">\/</a> <font color="Maroon" title="c6">q'</font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_6"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="lopclset.html#D2" target="_self" title="LOPCLSET:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:10_6"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:10"/><b>then </b>
(  <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> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="lopclset.html#E2:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="lopclset.html#E3:10"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="lopclset.html#E4:10"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="lopclset.html#E5:10"><i><font color="Green" title="E14">A9</font></i></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></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
 <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>
 <b>by </b><i><a class="ref" href="lattices.html#D10" title="LATTICES:def.10">LATTICES:def 10</a></i>;<br/>


</div>
