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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #);<br/>
<div><i><font color="Green" title="E46">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #);<br/><b>let </b><font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c8">B'</font> = <font color="Maroon" title="c4">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> ;<br/><b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c6">H2</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82_1"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="group_4.html#D12" target="_self" title="GROUP_4:def.12">Def12</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E47">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #);<br/><b>let </b><font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:82_2"/><i><font color="Green" title="E48">A4</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c8">B'</font> = <font color="Maroon" title="c4">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> ;<br/><b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c6">H2</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82_2"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="group_4.html#D11" target="_self" title="GROUP_4:def.11">Def11</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #);<br/><b>reconsider </b><font color="Maroon" title="c6">H1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">H2</font> = <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c8">H3</font> = <font color="Maroon" title="c5">C</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<font color="Maroon" title="c6">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H2</font>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">H2</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c6">H1</font>

<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>
;<br/><a NAME="E3:82_3"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c6">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c7">H2</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c8">H3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c8">H3</font>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H2</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c8">H3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T75" target="_self" title="GROUP_4:th.75">Th75</a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="group_4.html#E3:82_3"><i><font color="Green" title="E48">A5</font></i></a></i>
;<br/><br/><a NAME="E5:82_3"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H2</font>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="group_4.html#T84" target="_self" title="GROUP_4:th.84">Th84</a></i>
;<br/><br/><b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> = 
<font color="Maroon" title="c7">H2</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c6">H1</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a></i>
;<br/><a NAME="E8:82_3"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c7">H2</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c8">H3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c8">H3</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="group_4.html#E5:82_3"><i><font color="Green" title="E49">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H2</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c8">H3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T102" title="GROUP_2:th.102">GROUP_2:102</a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="group_4.html#E8:82_3"><i><font color="Green" title="E50">A7</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
<font color="Maroon" title="c6">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="group_4.html#E3:82_3"><i><font color="Green" title="E48">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="group_4.html#T85" target="_self" title="GROUP_4:th.85">Th85</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:82"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 
<b>by </b><i><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="E5:82"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</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/>

<b>reconsider </b><font color="Maroon" title="c3">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #) as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="txt" href="group_4.html#E4:82"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="lattices.html#D10" title="LATTICES:def.10">LATTICES:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">E</font> =  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c5">A</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">E</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">A</font> = 
<span class="p1">(<span class="default"><a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c6">H</font>
<b>by </b><i><a class="txt" href="group_4.html#E1:82"><i><font color="Green" title="E46">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">E</font>
<b>by </b><i><a class="ref" href="group_2.html#T103" title="GROUP_2:th.103">GROUP_2:103</a></i>
;<br/><b>hence </b><a NAME="E3:82_4"/>
<font color="Maroon" title="c5">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:82"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">F</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c6">A</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">F</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">A</font> = 
<span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">H</font>
<b>by </b><i><a class="txt" href="group_4.html#E2:82"><i><font color="Green" title="E47">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font>
<b>by </b><i><a class="ref" href="group_4.html#T77" target="_self" title="GROUP_4:th.77">Th77</a></i>
;<br/><b>hence </b><a NAME="E3:82_5"/>
<font color="Maroon" title="c6">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:82"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></i>;<br/>


</div>
