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

<b>set </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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> #);<br/>
<a NAME="E1:69_1_1"/><i><font color="Green" title="E51">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E1:69_1_1_1"/>
 <a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T30" target="_self" title="MSUALG_2:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1: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="E2:69_1_1"/><i><font color="Green" title="E52">A2</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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font></span><br/>

<a NAME="E1:69_1_1_2"/>
 <a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T31" target="_self" title="MSUALG_2:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</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="E3:69_1_1"/><i><font color="Green" title="E53">A3</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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<a NAME="E2:69_1_1_3"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">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="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">U1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">U2</font> = <font color="Maroon" title="c7">y</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<a NAME="E2:69_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">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="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c7">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U2</font></span>)</span> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c9">U2</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D21" target="_self" title="MSUALG_2:def.21">Def21</a></i>
<br/>.= 
<font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="ref" href="msualg_2.html#T29" target="_self" title="MSUALG_2:th.29">Th29</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>hence </b><a NAME="E3:69_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <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="E4:69_1_1"/><i><font color="Green" title="E54">A4</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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E1:69_1_1_4"/>
 <a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T32" target="_self" title="MSUALG_2:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_4"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1: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="E5:69_1_1"/><i><font color="Green" title="E55">A5</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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font></span><br/>

<a NAME="E1:69_1_1_5"/>
 <a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T33" target="_self" title="MSUALG_2:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_5"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</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="E6:69_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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="69_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<a NAME="E2:69_1_1_6"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">U1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">U2</font> = <font color="Maroon" title="c7">y</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<a NAME="E2:69_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c9">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D21" target="_self" title="MSUALG_2:def.21">Def21</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span> = 
<font color="Maroon" title="c8">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">U1</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c9">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="msualg_2.html#T28" target="_self" title="MSUALG_2:th.28">Th28</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>hence </b><a NAME="E3:69_1_1_6"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <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="E7:69_1_1"/><b>then </b>
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> #) is <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a> &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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="msualg_2.html#E1:69_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="msualg_2.html#E2:69_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="msualg_2.html#E3:69_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="msualg_2.html#E4:69_1_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="msualg_2.html#E5:69_1_1"><i><font color="Green" title="E55">A5</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="E8:69_1_1"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" title="MSUALG_2:func.15">MSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">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>
