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

<span class="kw">set </span><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="E1:48_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T25" target="_self" onmouseover="rs('unialg_2/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="binop_1.html#D3" onmouseover="rs('binop_1/D3')" onmouseout="rh()">BINOP_1:def 3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:48_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_2"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T26" target="_self" onmouseover="rs('unialg_2/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="binop_1.html#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:48_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_3"/><span class="lab"><font color="Green" title="E33">A6</font></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="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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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#V1" title="UNIALG_1:attr.1">strict</a>   <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D14" target="_self" onmouseover="rs('unialg_2/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E2:48_1_1_3_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<span class="kw">hence </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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D16" target="_self" onmouseover="rs('unialg_2/D16')" onmouseout="rh()">Def16</a></span>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T22" target="_self" onmouseover="rs('unialg_2/T22')" onmouseout="rh()">Th22</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_3"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:48_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_4"/><span class="lab"><font color="Green" title="E34">A8</font></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> <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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#V1" title="UNIALG_1:attr.1">strict</a>   <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D14" target="_self" onmouseover="rs('unialg_2/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E2:48_1_1_4_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D16" target="_self" onmouseover="rs('unialg_2/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<span class="kw">hence </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> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T23" target="_self" onmouseover="rs('unialg_2/T23')" onmouseout="rh()">Th23</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_4"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:48_1_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_5"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T27" target="_self" onmouseover="rs('unialg_2/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_5"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="binop_1.html#D3" onmouseover="rs('binop_1/D3')" onmouseout="rh()">BINOP_1:def 3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" onmouseover="rs('lattices/D2')" onmouseout="rh()">LATTICES:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:48_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#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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="E2:48_1_1_6"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T24" target="_self" onmouseover="rs('unialg_2/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_6"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="binop_1.html#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" onmouseover="rs('lattices/D1')" onmouseout="rh()">LATTICES:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:48_1_1"/><span class="kw">then </span>
(  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="lattices.html#D4" onmouseover="rs('lattices/D4')" onmouseout="rh()">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" onmouseover="rs('lattices/D5')" onmouseout="rh()">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" onmouseover="rs('lattices/D6')" onmouseout="rh()">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" onmouseover="rs('lattices/D7')" onmouseout="rh()">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" onmouseover="rs('lattices/D8')" onmouseout="rh()">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" onmouseover="rs('lattices/D9')" onmouseout="rh()">LATTICES:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
