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

<b>set </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>
<a NAME="E1:81_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:81_1_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="osalg_2.html#D4" target="_self">Def4</a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:81_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">c</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:81_1_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:81_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_3"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="txt" href="osalg_2.html#E3:81_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E2:81_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="msualg_2.html#K11">/\</a> <font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="msualg_2.html#K11">/\</a> <font color="Maroon">U2</font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font>
<b>by </b><i><a class="txt" href="osalg_2.html#E4:81_1_1"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:81_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:81_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_4"/><i><font color="Green">E73</font></i>: 
 <a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:81_1_1_4"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:81_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">c</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_5"/><i><font color="Green">E75</font></i>: 
 <a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:81_1_1_5"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:81_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #), <font color="Maroon">b</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> ;<br/>
<a NAME="E2:81_1_1_6"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="txt" href="osalg_2.html#E3:81_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E2:81_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E4:81_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">U1</font> <a href="msualg_2.html#K11">/\</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="osalg_2.html#E2:81_1_1"><i><font color="Green">E44</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:81_1_1_6"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:81_1_1"/><b>then </b>
(  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) is <a href="lattices.html#V3">strict</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</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="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="ref" href="osalg_2.html#T11" target="_self">Th11</a>, <a class="ref" href="osalg_2.html#D4" target="_self">Def4</a>, , , , <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="E8:81_1_1"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="osalg_2.html#K15" target="_self">OSAlg_meet</a> <font color="Maroon">U0</font></span>)</span> #) is  <a href="lattices.html#V3">strict</a> <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>
