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

<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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #);<br/>
<a NAME="E1:53_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</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="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">y</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T27" target="_self" title="UNIALG_2:th.27">Th27</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> = 
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>
<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="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1"/><i><font color="Green" title="E37">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_2"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T28" target="_self" title="UNIALG_2:th.28">Th28</a></i>;<br/>
<b>thus </b><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> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<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="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_2"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<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>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">y</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_3"/><i><font color="Green" title="E39">A6</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c7">U1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">U2</font> = <font color="Maroon" title="c6">y</font> as  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D15" target="_self" title="UNIALG_2:def.15">Def15</a></i>;<br/>
<a NAME="E2:53_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c6">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U2</font></span>)</span> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U2</font>
<b>by </b><i><a class="ref" href="unialg_2.html#D16" target="_self" title="UNIALG_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T26" target="_self" title="UNIALG_2:th.26">Th26</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <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>,<font color="Maroon" title="c4">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="c4">b</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_3"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:53_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</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="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">y</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_4"/><i><font color="Green" title="E40">A8</font></i>: 
 <a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T29" target="_self" title="UNIALG_2:th.29">Th29</a></i>;<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="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_4"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</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="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:53_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_5"/><i><font color="Green" title="E41">A10</font></i>: 
 <a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T30" target="_self" title="UNIALG_2:th.30">Th30</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="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<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="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_5"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<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>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:53_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_6"><b>proof </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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">y</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<a NAME="E2:53_1_1_6"/><i><font color="Green" title="E41">A11</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c7">U1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">U2</font> = <font color="Maroon" title="c6">y</font> as  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D15" target="_self" title="UNIALG_2:def.15">Def15</a></i>;<br/>
<a NAME="E2:53_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D16" target="_self" title="UNIALG_2:def.16">Def16</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> = 
<font color="Maroon" title="c7">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T25" target="_self" title="UNIALG_2:th.25">Th25</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<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>
<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="c3">a</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:53_1_1_6"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:53_1_1"/><b>then </b>
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</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="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:53_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="unialg_2.html#E2:53_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="unialg_2.html#E3:53_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="unialg_2.html#E4:53_1_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="unialg_2.html#E5:53_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> #) is  <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a> <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
