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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a></span><br/>

<b>set </b><font color="Maroon" title="c2">S</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #);<br/>
<a NAME="E1:68"/><i><font color="Green" title="E52">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c5">W1</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:68"/><i><font color="Green" title="E53">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c8">W3</font> = <font color="Maroon" title="c5">C</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">AB</font> = <font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c7">W2</font>, <font color="Maroon" title="c10">BC</font> = <font color="Maroon" title="c7">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c8">W3</font> as   <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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c10">BC</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c8">W3</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c8">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self" title="RUSUB_2:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">AB</font>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.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>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:68"/><i><font color="Green" title="E54">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">AB</font> = <font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c6">W2</font> as   <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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.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>,<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">AB</font>,<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E30"><i><font color="Green" title="E23">Lm6</font></i></a>, <a class="ref" href="rusub_1.html#T24" title="RUSUB_1:th.24">RUSUB_1:24</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:68"/><i><font color="Green" title="E55">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c6">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c5">W1</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T14" target="_self" title="RUSUB_2:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:68"/><i><font color="Green" title="E56">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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c8">W3</font> = <font color="Maroon" title="c5">C</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">AB</font> = <font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c7">W2</font>, <font color="Maroon" title="c10">BC</font> = <font color="Maroon" title="c7">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W3</font> as   <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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c10">BC</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W2</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W3</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T15" target="_self" title="RUSUB_2:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">AB</font>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.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>,<font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:68"/>
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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="68_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">AB</font> = <font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font> as   <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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">AB</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E32"><i><font color="Green" title="E24">Lm7</font></i></a>, <a class="ref" href="rusub_1.html#T24" title="RUSUB_1:th.24">RUSUB_1:24</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:68"/><b>then </b>
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:68"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E56">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="E8:68"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
