<?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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>
<a NAME="E1:10"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<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="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">p'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">r'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">l'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> )
 ;<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="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">k'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p'</font> <a href="lopclset.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">r'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">q'</font></span>)</span> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">l'</font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">r'</font>
<b>by </b><i>, </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>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> )
 ;<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> = 
<font color="Maroon">k'</font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">q'</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">q'</font></span>)</span> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">q'</font>
<b>by </b><i>, </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>
<a NAME="E4:10"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<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="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">p'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:10"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">q</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">r'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">l'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_5"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> )
 ;<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="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">k'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p'</font> <a href="lopclset.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">r'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">q'</font></span>)</span> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">l'</font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">r'</font>
<b>by </b><i>, </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>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:10"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #) holds  <font color="Olive">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_meet</a> <font color="Maroon">T</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font>, <font color="Maroon">k'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:10_6"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> )
 ;<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="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">k'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p'</font> <a href="lopclset.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">q'</font></span>)</span>
<b>by </b><i>, </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="E7:10"/><b>then </b>
(  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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="E8:10"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K4" target="_self">T_join</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopclset.html#K5" target="_self">T_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>
