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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<font color="Maroon">q</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">p</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T22">XBOOLE_1:22</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font> = 
<font color="Maroon">q</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #), <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #);<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> = 
<font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:8"/><b>then </b>
(  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V4">join-commutative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V5">join-associative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V8">meet-absorbing</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V6">meet-commutative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V7">meet-associative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i>, , , , , <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K1" target="_self">Topology_of</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K4" target="_self">Top_Union</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="openlatt.html#K5" target="_self">Top_Meet</a> <font color="Maroon">T</font></span>)</span> #) is   <a href="lattices.html#NM1">Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>


</div>
