### Boolean Properties of Lattices

by
Agnieszka Julia Marasik

Received March 28, 1994

Copyright (c) 1994 Association of Mizar Users

MML identifier: BOOLEALG
[ MML identifier index ]

```environ

vocabulary LATTICES, BOOLE, SUBSET_1;
notation STRUCT_0, LATTICES;
constructors LATTICES, XBOOLE_0;
clusters LATTICES, ZFMISC_1, XBOOLE_0;
theorems LATTICES, FILTER_0, LATTICE4;

begin :: General Lattices

reserve L for Lattice;
reserve X,Y,Z,V for Element of L;

definition let L,X,Y;
func X \ Y -> Element of L equals :Def1:
X "/\" Y`;
correctness;
end;

definition let L,X,Y;
func X \+\ Y -> Element of L equals
:Def2:  (X \ Y) "\/" (Y \ X);
correctness;
end;

definition let L,X,Y;
redefine pred X = Y means :Def3:
X [= Y & Y [= X;
compatibility by LATTICES:26;
end;

definition let L,X,Y;
pred X meets Y means
:Def4: X "/\" Y <> Bottom L;
antonym X misses Y;
end;

Lm1:  X [= Z & Y [= Z implies X "\/" Y [= Z
proof
assume that A1: X [= Z and
A2: Y [= Z;
A3:  X "\/" Y [= Z "\/" Y by A1,FILTER_0:1;
Z "\/" Y [= Z "\/" Z by A2,FILTER_0:1;
then Z "\/" Y [= Z by LATTICES:17;
hence thesis by A3,LATTICES:25;
end;

canceled 2;

theorem
X "\/" Y [= Z implies X [= Z & Y [= Z
proof
assume A1: X "\/" Y [= Z;
then X "/\" (X "\/" Y) [= X "/\" Z by LATTICES:27;
then A2:  X [= X "/\" Z by LATTICES:def 9;
A3:X "/\" Z [= Z by LATTICES:23;
Y "/\" (Y "\/" X) [= Y "/\" Z by A1,LATTICES:27;
then A4:  Y [= Y "/\" Z by LATTICES:def 9;
Y "/\" Z [= Z by LATTICES:23;
hence thesis by A2,A3,A4,LATTICES:25;
end;

theorem
X "/\" Y [= X "\/" Z
proof
X "/\" Y [= X & X [= X "\/" Z by LATTICES:22,23;
hence thesis by LATTICES:25;
end;

canceled;

theorem
X [= Z implies X \ Y [= Z
proof
assume X [= Z;
then A1: X "/\" Y` [= Z "/\" Y` by LATTICES:27;
Z "/\" Y` [= Z by LATTICES:23;
then X "/\" Y` [= Z by A1,LATTICES:25;
hence thesis by Def1;
end;

theorem
Th7: X [= Y implies X \ Z [= Y \ Z
proof
assume A1: X [= Y;
X "/\" Z` = X \ Z & Y "/\" Z` = Y \ Z by Def1;
hence thesis by A1,LATTICES:27;
end;

theorem
Th8: X \ Y [= X
proof
X "/\" Y` [= X by LATTICES:23;
hence thesis by Def1;
end;

theorem
X \ Y [= X \+\ Y
proof X \+\ Y = (X \ Y) "\/" (Y \ X) by Def2;
hence thesis by LATTICES:22;
end;

theorem
X \ Y [= Z & Y \ X [= Z implies X \+\ Y [= Z
proof
assume that A1: X \ Y [= Z and
A2: Y \ X [= Z;
(X \ Y) "\/" (Y \ X) [= Z by A1,A2,Lm1;
hence thesis by Def2;
end;

theorem
X = Y "\/" Z iff Y [= X & Z [= X & for V st Y [= V & Z [= V
holds X [= V
proof
thus X = Y "\/" Z implies
Y [= X & Z [= X & for V st Y [= V & Z [= V holds X [= V
by Lm1,LATTICES:22;
assume that A1: Y [= X and
A2: Z [= X and
A3: for V st Y [= V & Z [= V holds X [= V;
Y [= Y "\/" Z & Z [= Y "\/" Z by LATTICES:22;
then A4:  X [= Y "\/" Z by A3;
Y "\/" Z [= X by A1,A2,Lm1;
hence thesis by A4,LATTICES:26;
end;

theorem
X = Y "/\" Z iff X [= Y & X [= Z & for V st V [= Y & V [= Z
holds V [= X
proof
thus X = Y "/\" Z implies
X [= Y & X [= Z & for V st V [= Y & V [= Z holds V [= X by FILTER_0:7,
LATTICES:23;
assume that A1: X [= Y & X [= Z and
A2: for V st V [= Y & V [= Z holds V [= X;
A3:  X [= Y "/\" Z by A1,FILTER_0:7;
Y "/\" Z [= Y & Y "/\" Z [= Z by LATTICES:23;
then Y "/\" Z [= X by A2;
hence X = Y "/\" Z by A3,Def3;
end;

canceled;

theorem
Th14: X "/\" (Y \ Z) = (X "/\" Y) \ Z
proof
X "/\" (Y \ Z)
= X "/\" (Y "/\" Z`) by Def1
.= (X "/\" Y) "/\" Z` by LATTICES:def 7
.= (X "/\" Y) \ Z by Def1;
hence thesis;
end;

theorem Th15:
X meets Y implies Y meets X
proof
assume X meets Y;
then X "/\" Y <> Bottom L by Def4;
hence thesis by Def4;
end;

theorem
X meets X iff X <> Bottom L
proof
hereby assume X meets X;
then X "/\" X <> Bottom L by Def4;
hence X <> Bottom L by LATTICES:18;
end;
assume A1: X <> Bottom L;
X "/\" X = X by LATTICES:18;
hence thesis by A1,Def4;
end;

theorem Th17:
X \+\ Y = Y \+\ X
proof
thus X \+\ Y = (X \ Y) "\/" (Y \ X) by Def2
.= Y \+\ X by Def2;
end;

definition let L, X, Y;
redefine pred X meets Y;
symmetry by Th15;
redefine func X \+\ Y;
commutativity by Th17;
end;

begin :: Modular Lattices

reserve L for M_Lattice;
reserve X,Y for Element of L;

canceled 3;

theorem
X misses Y implies Y misses X
proof
assume X misses Y;
then Y "/\" X = Bottom L by Def4;
hence thesis by Def4;
end;

begin   ::Distributive Lattices

reserve L for D_Lattice;
reserve X,Y,Z for Element of L;

theorem
(X "/\" Y) "\/" (X "/\" Z) = X implies X [= Y "\/" Z
proof
assume (X "/\" Y) "\/" (X "/\" Z) = X;
then X "/\" (Y "\/" Z) = X by LATTICES:def 11;
hence thesis by LATTICES:21;
end;

canceled;

theorem
Th24: (X "\/" Y) \ Z = (X \ Z) "\/" (Y \ Z)
proof
thus (X "\/" Y) \ Z
= (X "\/" Y) "/\" Z` by Def1
.=(X "/\" Z`) "\/" (Y "/\" Z`) by LATTICES:def 11
.= (X \ Z) "\/" (Y "/\" Z`) by Def1
.= (X \ Z) "\/" (Y \ Z) by Def1;
end;

begin   ::Distributive bounded Lattices

reserve L for 0_Lattice;
reserve X,Y,Z for Element of L;

theorem
Th25: X [= Bottom L implies X = Bottom L
proof
assume A1: X [= Bottom L;
Bottom L [= X by LATTICES:41;
hence thesis by A1,LATTICES:26;
end;

theorem
X [= Y & X [= Z & Y "/\" Z = Bottom L implies X = Bottom L
proof
assume that A1: X [= Y and
A2: X [= Z and
A3: Y "/\" Z = Bottom L;
X [= Bottom L by A1,A2,A3,FILTER_0:7;
hence X = Bottom L by Th25;
end;

theorem
Th27: X "\/" Y = Bottom L iff X = Bottom L & Y = Bottom L
proof
thus X "\/" Y = Bottom L implies X = Bottom L & Y = Bottom L
proof
assume A1: X "\/" Y = Bottom L;
then A2: X [= Bottom L & Bottom L [= X by LATTICES:22,41;
Y [= Bottom L & Bottom L [= Y by A1,LATTICES:22,41;
hence thesis by A2,Def3;
end;
thus X = Bottom L & Y = Bottom L implies X "\/" Y = Bottom L by LATTICES:39
;
end;

theorem
X [= Y & Y "/\" Z = Bottom L implies X "/\" Z = Bottom L
proof
assume X [= Y;
then X "/\" Z [= Y "/\" Z by LATTICES:27;
hence thesis by Th25;
end;

theorem
Th29:  Bottom L \ X = Bottom L
proof
Bottom L \ X = Bottom L "/\" X` by Def1
.= Bottom L by LATTICES:40;
hence thesis;
end;

theorem
Th30:  X meets Y & Y [= Z implies X meets Z
proof
assume that A1: X meets Y and
A2: Y [= Z;
A3: X "/\" Y <> Bottom L by A1,Def4;
X "/\" Y [= X "/\" Z by A2,LATTICES:27;
then X "/\" Z <> Bottom L by A3,Th25;
hence thesis by Def4;
end;

theorem
Th31:  X meets Y "/\" Z implies X meets Y & X meets Z
proof
assume X meets Y "/\" Z;
then A1: X "/\" (Y "/\" Z) <> Bottom L by Def4;
then A2:  X "/\" Y "/\" Z <> Bottom L by LATTICES:def 7;
X "/\" Z "/\" Y <> Bottom L by A1,LATTICES:def 7;
then X "/\" Y <> Bottom L & X "/\" Z <> Bottom L by A2,LATTICES:40;
hence thesis by Def4;
end;

theorem
X meets Y \ Z implies X meets Y
proof
assume X meets Y \ Z;
then X "/\" (Y \ Z) <> Bottom L by Def4;
then X "/\" (Y "/\" Z`) <> Bottom L by Def1;
then X "/\" Y "/\" Z` <> Bottom L by LATTICES:def 7;
then X "/\" Y <> Bottom L by LATTICES:40;
hence thesis by Def4;
end;

theorem
X misses Bottom L
proof
assume X meets Bottom L;
then X "/\" Bottom L <> Bottom L by Def4;
hence contradiction by LATTICES:40;
end;

theorem
X misses Z & Y [= Z implies X misses Y by Th30;

theorem
X misses Y or X misses Z implies X misses Y "/\" Z by Th31;

theorem
X [= Y & X [= Z & Y misses Z implies X = Bottom L
proof
assume X [= Y & X [= Z & Y misses Z;
then X [= Y "/\" Z & Y "/\" Z = Bottom L by Def4,FILTER_0:7;
hence thesis by Th25;
end;

theorem
X misses Y implies (Z "/\" X) misses (Z "/\" Y)
& (X "/\" Z) misses (Y "/\" Z)
proof
assume A1: X misses Y;
thus (Z "/\" X) misses (Z "/\" Y)
proof
(Z "/\" X) "/\" (Z "/\" Y) = Z "/\" (X "/\" (Y "/\" Z)) by LATTICES:def 7
.= Z "/\" ((X "/\" Y) "/\" Z) by LATTICES:def 7
.= Z "/\" (Bottom L "/\" Z) by A1,Def4
.= Z "/\" Bottom L by LATTICES:40
.= Bottom L by LATTICES:40;
hence thesis by Def4;
end;
hence thesis;
end;

begin   ::Boolean Lattices

reserve L for B_Lattice;
reserve X,Y,Z,V for Element of L;

theorem
X \ Y [= Z implies X [= Y "\/" Z
proof
assume X \ Y [= Z;
then X "/\" Y` [= Z by Def1;
then Y "\/" (X "/\" Y`) [= Y "\/" Z by FILTER_0:1;
then (Y "\/" X) "/\" (Y "\/" Y`) [= Y "\/" Z by LATTICES:31;
then (Y "\/" X) "/\" Top L [= Y "\/" Z by LATTICES:48;
then A1: X "\/" Y [= Y "\/" Z by LATTICES:43;
X [= X "\/" Y by LATTICES:22;
hence thesis by A1,LATTICES:25;
end;

theorem
Th39: X [= Y implies Z \ Y [= Z \ X
proof
assume X [= Y;
then A1: Y` [= X` by LATTICES:53;
Z \ Y = Z "/\" Y` & Z "/\" X` = Z \ X by Def1;
hence thesis by A1,LATTICES:27;
end;

theorem
X [= Y & Z [= V implies X \ V [= Y \ Z
proof
assume that A1: X [= Y and
A2: Z [= V;
A3: X \ V [= Y \ V by A1,Th7;
Y \ V [= Y \ Z by A2,Th39;
hence thesis by A3,LATTICES:25;
end;

theorem
X [= Y "\/" Z implies X \ Y [= Z & X \ Z [= Y
proof
assume A1: X [= Y "\/" Z;
thus X \ Y [= Z
proof
X "/\" Y` [= (Y "\/" Z) "/\" Y` by A1,LATTICES:27;
then X "/\" Y` [= (Y "/\" Y`) "\/" (Z "/\" Y`) by LATTICES:def 11;
then X \ Y [= (Y "/\" Y` ) "\/" (Z "/\" Y`) by Def1;
then X \ Y [= Bottom L "\/" (Z "/\" Y`) by LATTICES:47;
then A2: X \ Y [= Z "/\" Y` by LATTICES:39;
Z "/\" Y` [= Z by LATTICES:23;
hence thesis by A2,LATTICES:25;
end;
X "/\" Z` [= (Y "\/" Z) "/\" Z` by A1,LATTICES:27;
then X "/\" Z` [= (Y "/\" Z`) "\/" (Z "/\" Z`) by LATTICES:def 11;
then X \ Z [= (Y "/\" Z`) "\/" (Z "/\" Z`) by Def1;
then X \ Z [= (Y "/\" Z`) "\/" Bottom L by LATTICES:47;
then A3: X \ Z [= Y "/\" Z` by LATTICES:39;
Y "/\" Z` [= Y by LATTICES:23;
hence X \ Z [= Y by A3,LATTICES:25;
end;

theorem
X` [= (X "/\" Y)` & Y` [= (X "/\" Y)`
proof
A1:X` [= X` "\/" Y` by LATTICES:22;
Y` [= X` "\/" Y` by LATTICES:22;
hence thesis by A1,LATTICES:50;
end;

theorem
(X "\/" Y)` [= X` & (X "\/" Y)` [= Y`
proof
thus (X "\/" Y)` [= X`
proof
(X "\/" Y)` = X` "/\" Y` by LATTICES:51;
hence thesis by LATTICES:23;
end;
(X "\/" Y)` = Y` "/\" X` by LATTICES:51;
hence thesis by LATTICES:23;
end;

theorem
X [= Y \ X implies X = Bottom L
proof
assume X [= Y \ X;
then A1: X [= Y "/\" X` by Def1;
X "/\" (Y "/\" X`)
= Y "/\" (X` "/\" X) by LATTICES:def 7
.= Y "/\" Bottom L by LATTICES:47
.= Bottom L by LATTICES:40;
hence thesis by A1,LATTICES:21;
end;

theorem
X [= Y implies Y = X "\/" (Y \ X)
proof
assume A1: X [= Y;
Y = Y "/\" Top L by LATTICES:43
.= Y "/\" ( X "\/" X`) by LATTICES:48
.= (Y "/\" X) "\/" ( Y "/\" X`) by LATTICES:def 11
.= X "\/" (Y "/\" X`) by A1,LATTICES:21
.= X "\/" (Y \ X) by Def1;
hence thesis;
end;

theorem
Th46: X \ Y = Bottom L iff X [= Y
proof
thus X \ Y = Bottom L implies X [= Y
proof
assume
A1: X \ Y = Bottom L;
X "/\" Y` = Bottom L implies X [= Y`` by LATTICES:52;
hence thesis by A1,Def1,LATTICES:49;
end;
assume X [= Y;
then X "/\" Y` [= Y` "/\" Y by LATTICES:27;
then X "/\" Y` [= Bottom L by LATTICES:47;
then A2:  X \ Y [= Bottom L by Def1;
Bottom L [= X \ Y by LATTICES:41;
hence thesis by A2,Def3;
end;

theorem
X [= (Y "\/" Z) & X "/\" Z = Bottom L implies X [= Y
proof
assume that A1: X [= (Y "\/" Z) and
A2: X "/\" Z = Bottom L;
X "/\" (Y "\/" Z) = X by A1,LATTICES:21;
then (X "/\" Y) "\/" Bottom L = X by A2,LATTICES:def 11;
then X "/\" Y = X by LATTICES:39;
hence thesis by LATTICES:21;
end;

theorem
X "\/" Y = (X \ Y) "\/" Y
proof
thus X "\/" Y = (X "\/" Y) "/\" Top L by LATTICES:43
.= (X "\/" Y) "/\" (Y` "\/" Y) by LATTICES:48
.= (X "/\" Y`) "\/" Y by LATTICES:31
.= (X \ Y) "\/" Y by Def1;
end;

theorem
X \ (X "\/" Y) = Bottom L
proof
X [= X "\/" Y & X [= Y "\/" X by LATTICES:22;
hence thesis by Th46;
end;

theorem
Th50: X \ X "/\" Y = X \ Y & X \ Y "/\" X = X \ Y
proof
thus X \ X "/\" Y = X \ Y
proof
X \ X "/\" Y = X "/\" (X "/\" Y)` by Def1
.= X "/\" (X` "\/" Y`) by LATTICES:50
.= (X "/\" X`) "\/" (X "/\" Y`) by LATTICES:def 11
.= Bottom L "\/" (X "/\" Y`) by LATTICES:47
.= X "/\" Y` by LATTICES:39;
hence thesis by Def1;
end;
X \ Y "/\" X = X "/\" (Y "/\" X)` by Def1
.= X "/\" (Y` "\/" X`) by LATTICES:50
.= (X "/\" Y`) "\/" (X "/\" X`) by LATTICES:def 11
.= (X "/\" Y`) "\/" Bottom L by LATTICES:47
.= X "/\" Y` by LATTICES:39;
hence thesis by Def1;
end;

theorem
(X \ Y) "/\" Y = Bottom L
proof
(X \ Y) "/\" Y = (X "/\" Y`) "/\" Y by Def1
.= X "/\" (Y`"/\" Y) by LATTICES:def 7
.= X "/\" Bottom L by LATTICES:47;
hence thesis by LATTICES:40;
end;

theorem
Th52: X "\/" (Y \ X) = X "\/" Y & (Y \ X) "\/" X = Y "\/" X
proof
thus X "\/" (Y \ X) = X "\/" Y
proof
X "\/" (Y \ X)
= X "\/" (Y "/\" X`) by Def1
.= (X "\/" Y) "/\" (X "\/" X`) by LATTICES:31
.= (X "\/" Y) "/\" Top L by LATTICES:48;
hence thesis by LATTICES:43;
end;
(Y \ X) "\/" X = (Y "/\" X`) "\/" X by Def1
.= (Y "\/" X) "/\" (X` "\/" X) by LATTICES:31
.= (Y "\/" X) "/\" Top L by LATTICES:48;
hence thesis by LATTICES:43;
end;

theorem
Th53: (X "/\" Y) "\/" (X \ Y) = X
proof
(X "/\" Y) "\/" (X \ Y) = (X "/\" Y) "\/" (X "/\" Y`) by Def1
.=((X "/\" Y) "\/" X) "/\" ((X "/\" Y) "\/" Y`) by LATTICES:31
.= X "/\" ((X "/\" Y) "\/" Y`) by LATTICES:def 8
.= X "/\" ((X "\/" Y`) "/\" (Y "\/" Y`)) by LATTICES:31
.= X "/\" ((X "\/" Y`) "/\" Top L) by LATTICES:48
.= X "/\" (X "\/" Y`) by LATTICES:43
.= X by LATTICES:def 9;
hence thesis;
end;

theorem
Th54: X \ (Y \ Z) = (X \ Y) "\/" (X "/\" Z)
proof
X \ (Y \ Z) = X \ (Y "/\" Z`) by Def1
.= X "/\" (Y "/\" Z`)` by Def1
.= X "/\" (Y`"\/" Z``) by LATTICES:50
.= X "/\" (Y`"\/" Z) by LATTICES:49
.= (X "/\" Y`) "\/" (X "/\" Z) by LATTICES:def 11;
hence thesis by Def1;
end;

theorem
X \ (X \ Y) = X "/\" Y
proof
X \ (X \ Y) = X "/\" (X \ Y)` by Def1
.= X "/\" ( X "/\" Y`)` by Def1
.= X "/\" (X` "\/" Y``) by LATTICES:50
.= X "/\" (X` "\/" Y) by LATTICES:49
.= (X "/\" X`) "\/" (X "/\" Y) by LATTICES:def 11
.= Bottom L "\/" (X "/\" Y) by LATTICES:47;
hence thesis by LATTICES:39;
end;

theorem
(X "\/" Y) \ Y = X \ Y
proof
(X "\/" Y) \ Y = (X "\/" Y) "/\" Y` by Def1
.= (X "/\" Y`) "\/" (Y "/\" Y`) by LATTICES:def 11
.= (X "/\" Y`) "\/" Bottom L by LATTICES:47
.= X "/\" Y` by LATTICES:39;
hence thesis by Def1;
end;

theorem
X "/\" Y = Bottom L iff X \ Y = X
proof
thus X "/\" Y = Bottom L implies X \ Y = X
proof
assume X "/\" Y = Bottom L;
then X [= Y` by LATTICES:52;
then X "/\" X [= X "/\" Y` by LATTICES:27;
then X [= X "/\" Y` by LATTICES:18;
then A1: X [= X \ Y by Def1;
X \ Y [= X by Th8;
hence thesis by A1,Def3;
end;
assume X \ Y = X;
then X "/\" Y` = X by Def1;
then X` "\/" Y`` = X` by LATTICES:50;
then X "/\" (X` "\/" Y) [= X "/\" X` by LATTICES:49;
then (X "/\" X`) "\/" (X "/\" Y) [= X "/\" X` by LATTICES:def 11;
then Bottom L "\/" (X "/\" Y) [= X "/\" X` by LATTICES:47;
then Bottom L "\/" (X "/\" Y) [= Bottom L by LATTICES:47;
then A2: X "/\" Y [= Bottom L by LATTICES:39;
Bottom L [= X "/\" Y by LATTICES:41;
hence thesis by A2,Def3;
end;

theorem
X \ (Y "\/" Z) = (X \ Y) "/\" (X \ Z)
proof
thus X \ (Y "\/" Z) = X "/\" (Y "\/" Z)` by Def1
.= X "/\" (Y` "/\" Z`) by LATTICES:51
.= (X "/\" X) "/\" (Y` "/\" Z`) by LATTICES:18
.= (X "/\" X "/\" Y`) "/\" Z` by LATTICES:def 7
.= (X "/\" (X "/\" Y`)) "/\" Z` by LATTICES:def 7
.= (X "/\" Y`) "/\" (X "/\" Z`) by LATTICES:def 7
.= (X \ Y) "/\" (X "/\" Z`) by Def1
.= (X \ Y) "/\" (X \ Z) by Def1;
end;

theorem
Th59: X \ (Y "/\" Z) = (X \ Y) "\/" (X \ Z)
proof
thus X \ (Y "/\" Z)
= X "/\" (Y "/\" Z)` by Def1
.= X "/\" (Y` "\/" Z`) by LATTICES:50
.=(X "/\" Y`) "\/" (X "/\" Z`) by LATTICES:def 11
.=(X \ Y) "\/" (X "/\" Z`) by Def1
.=(X \ Y) "\/" (X \ Z) by Def1;
end;

theorem
X "/\" (Y \ Z) = X "/\" Y \ X "/\" Z & (Y \ Z) "/\" X = Y "/\" X \ Z "/\" X
proof
A1: X "/\" Y [= X by LATTICES:23;
A2: X "/\" Y \ X "/\" Z = ((X "/\" Y) \ X) "\/" ((X "/\" Y) \ Z) by Th59
.= Bottom L "\/" ((X "/\" Y) \ Z) by A1,Th46
.= (X "/\" Y) \ Z by LATTICES:39;
hence X "/\" (Y \ Z) = X "/\" Y \ X "/\" Z by Th14;
thus (Y \ Z) "/\" X = Y "/\" X \ Z "/\" X by A2,Th14;
end;

theorem
Th61: (X "\/" Y) \ (X "/\" Y) = (X \ Y) "\/" (Y \ X)
proof
(X "\/" Y) \ (X "/\" Y)
= (X "\/" Y) "/\" (X "/\" Y)` by Def1
.= (X "\/" Y) "/\" (X` "\/" Y`) by LATTICES:50
.= ((X "\/" Y) "/\" X`) "\/" ((X "\/" Y) "/\" Y`) by LATTICES:def 11
.= ((X "/\" X`) "\/" (Y "/\" X`)) "\/" ((X "\/" Y) "/\"
Y`) by LATTICES:def 11
.= ((X "/\" X`) "\/" (Y "/\" X`)) "\/" ((X "/\" Y`) "\/" (Y "/\"
Y`)) by LATTICES:def 11
.= (Bottom L "\/" (Y "/\" X`)) "\/" ((X "/\" Y`) "\/" (Y "/\"
Y`)) by LATTICES:47
.= (Bottom L "\/" (Y "/\" X`)) "\/" ((X "/\" Y`) "\/" Bottom
L) by LATTICES:47
.= (Y "/\" X`) "\/" ((X "/\" Y`) "\/" Bottom L) by LATTICES:39
.= (Y "/\" X`) "\/" (X "/\" Y`) by LATTICES:39
.= (Y \ X) "\/" (X "/\" Y`) by Def1
.= (X \ Y) "\/" (Y \ X) by Def1;
hence thesis;
end;

theorem
Th62: (X \ Y) \ Z = X \ (Y "\/" Z)
proof
thus (X \ Y) \ Z = (X "/\" Y`) \ Z by Def1
.= (X "/\" Y`) "/\" Z` by Def1
.= X "/\" (Y` "/\" Z`) by LATTICES:def 7
.= X "/\" (Y "\/" Z)` by LATTICES:51
.= X \ (Y "\/" Z) by Def1;
end;

theorem
X \ Y = Y \ X implies X = Y
proof
assume A1: X \ Y = Y \ X;
X \ Y = X "/\" Y` by Def1;
then A2: X "/\" Y` = Y "/\" X` by A1,Def1;
then (X "/\" Y`) "/\" X = Y "/\" (X` "/\" X) by LATTICES:def 7
.= Y "/\" Bottom L by LATTICES:47;
then X "/\" (X "/\" Y`) = Bottom L by LATTICES:40;
then (X "/\" X) "/\" Y` = Bottom L by LATTICES:def 7;
then X "/\" Y` = Bottom L by LATTICES:18;
then (X "/\" Y`) "\/" X`= (X "/\" X`) "\/" X` by LATTICES:47;
then (X "/\" Y`) "\/" X`= X` by LATTICES:def 8;
then (X "\/" X`) "/\" (Y` "\/" X`) = X` by LATTICES:31;
then Top L "/\" (Y` "\/" X`) = X` by LATTICES:48;
then Y` "\/" X` = X` by LATTICES:43;
then Y` "/\" X` = Y` by LATTICES:def 9;
then Y` [= X` by LATTICES:21;
then X`` [= Y`` by LATTICES:53;
then X [= Y`` by LATTICES:49;
then A3: X [= Y by LATTICES:49;
X "/\" (Y` "/\" Y) = (Y "/\" X`) "/\" Y by A2,LATTICES:def 7;
then X "/\" Bottom L = (Y "/\" X`) "/\" Y by LATTICES:47;
then Bottom L = (X` "/\" Y) "/\" Y by LATTICES:40
.= X` "/\" (Y "/\" Y) by LATTICES:def 7
.= X` "/\" Y by LATTICES:18;
then X` [= Y` by LATTICES:52;
then Y`` [= X`` by LATTICES:53;
then Y [= X`` by LATTICES:49;
then Y [= X by LATTICES:49;
hence thesis by A3,Def3;
end;

canceled 2;

theorem
Th66: X \ X = Bottom L
proof
X \ X = X "/\" X` by Def1
.= Bottom L by LATTICES:47;
hence thesis;
end;

theorem
Th67: X \ Bottom L = X
proof
X \ Bottom L = X "/\" (Bottom L)` by Def1
.= X "/\" Top L by LATTICE4:37
.= X by LATTICES:43;
hence thesis;
end;

theorem
(X \ Y)` = X` "\/" Y
proof
(X \ Y)` = (X "/\" Y`)` by Def1
.= X` "\/" Y`` by LATTICES:50;
hence thesis by LATTICES:49;
end;

theorem
Th69: X meets Y "\/" Z iff X meets Y or X meets Z
proof
thus X meets Y "\/" Z implies X meets Y or X meets Z
proof
assume X meets Y "\/" Z;
then X "/\" (Y "\/" Z) <> Bottom L by Def4;
then (X "/\" Y) "\/" (X "/\" Z) <> Bottom L by LATTICES:def 11;
then (X "/\" Y) <> Bottom L or (X "/\" Z) <> Bottom L by LATTICES:39;
hence thesis by Def4;
end;
assume A1: X meets Y or X meets Z;
A2: X meets Y implies X meets Y "\/" Z
proof
assume A3: X "/\" Y <> Bottom L;
X "/\" Y [= (X "/\" Y) "\/" (X "/\" Z) by LATTICES:22;
then X "/\" Y [= X "/\" (Y "\/" Z) by LATTICES:def 11;
then X "/\" (Y "\/" Z) <> Bottom L by A3,Th25;
hence thesis by Def4;
end;
X meets Z implies X meets Y "\/" Z
proof
assume A4: X "/\" Z <> Bottom L;
(X "/\" Z) "\/" (X "/\" Y) = X "/\" (Y "\/" Z) by LATTICES:def 11;
then X "/\" (Y "\/" Z) <> Bottom L by A4,Th27;
hence thesis by Def4;
end;
hence thesis by A1,A2;
end;

theorem
Th70: X "/\" Y misses X \ Y
proof
(X "/\" Y) "/\" (X \ Y) = (X "/\" Y) "/\" (X "/\" Y`) by Def1
.= (X "/\" Y "/\" Y`) "/\" X by LATTICES:def 7
.= (X "/\" (Y "/\" Y`)) "/\" X by LATTICES:def 7
.= (X "/\" Bottom L) "/\" X by LATTICES:47
.= Bottom L "/\" X by LATTICES:40
.= Bottom L by LATTICES:40;
hence thesis by Def4;
end;

theorem
X misses Y "\/" Z iff X misses Y & X misses Z by Th69;

theorem
(X \ Y) misses Y
proof
(X \ Y) "/\" Y = (X "/\" Y`) "/\" Y by Def1
.= X "/\" (Y` "/\" Y) by LATTICES:def 7
.= X "/\" Bottom L by LATTICES:47
.= Bottom L by LATTICES:40;
hence thesis by Def4;
end;

theorem
X misses Y implies (X "\/" Y) \ Y = X & (X "\/" Y) \ X = Y
proof
assume A1: X "/\" Y = Bottom L;
thus (X "\/" Y) \ Y = X
proof
A2:  (X "\/" Y) \ Y = (X "\/" Y) "/\" Y` by Def1
.= (X "/\" Y`) "\/" (Y "/\" Y`) by LATTICES:def 11
.= (X "/\" Y`) "\/" Bottom L by LATTICES:47
.= X "/\" Y` by LATTICES:39;
X` "\/" (X "/\" Y) = X` by A1,LATTICES:39;
then (X` "\/" X) "/\" (X` "\/" Y) = X` by LATTICES:31;
then Top L "/\" (X` "\/" Y) = X` by LATTICES:48;
then (X` "\/" Y)` = X`` by LATTICES:43;
then (X` "\/" Y)` = X by LATTICES:49;
then X`` "/\" Y` = X by LATTICES:51;
hence thesis by A2,LATTICES:49;
end;
A3:  (X "\/" Y) \ X = (X "\/" Y) "/\" X` by Def1
.= (X "/\" X`) "\/" (Y "/\" X`) by LATTICES:def 11
.= Bottom L "\/" (Y"/\" X`) by LATTICES:47
.= Y "/\" X` by LATTICES:39;
Y` "\/" (X "/\" Y) = Y` by A1,LATTICES:39;
then (Y` "\/" X) "/\" (Y` "\/" Y) = Y` by LATTICES:31;
then (Y` "\/" X) "/\" Top L = Y` by LATTICES:48;
then (Y` "\/" X)` = Y`` by LATTICES:43;
then (Y` "\/" X)` = Y by LATTICES:49;
then Y`` "/\" X` = Y by LATTICES:51;
hence thesis by A3,LATTICES:49;
end;

theorem
X` "\/" Y` = X "\/" Y & X misses X` & Y misses Y` implies
X = Y` & Y = X`
proof
assume that A1: X` "\/" Y` = X "\/" Y and
A2: X misses X` and
A3: Y misses Y`;
A4: X "/\" X`= Bottom L by A2,Def4;
A5: Y "/\" Y` = Bottom L by A3,Def4;
(X "/\" X`) "\/" (X "/\" Y`) = X "/\" (X "\/" Y) by A1,LATTICES:def 11;
then X "/\" Y` = X "/\" (X "\/" Y) by A4,LATTICES:39;
then A6: X "/\" Y` = X by LATTICES:def 9;
X` "/\" (X` "\/" Y`) = Bottom L "\/" (X` "/\" Y) by A1,A4,LATTICES:def 11;
then A7: X` "/\" (X` "\/" Y`) = X` "/\" Y by LATTICES:39;
(Y "/\" X`) "\/" (Y "/\" Y`) = Y "/\" (X "\/" Y) by A1,LATTICES:def 11;
then Y "/\" X` = Y "/\" (X "\/" Y) by A5,LATTICES:39;
then A8: Y "/\" X` = Y by LATTICES:def 9;
Y` "/\" (X` "\/" Y`) = (Y` "/\" X) "\/" Bottom L by A1,A5,LATTICES:def 11;
then Y` "/\" (X` "\/" Y`) = Y` "/\" X by LATTICES:39;
hence X = Y` by A6,LATTICES:def 9;
thus Y = X` by A7,A8,LATTICES:def 9;
thus thesis;
end;

theorem
X` "\/" Y` = X "\/" Y & Y misses X` & X misses Y`
implies X = X` & Y = Y`
proof
assume that A1:  X` "\/" Y` = X "\/" Y and
A2:  Y misses X` and
A3:  X misses Y`;
A4: Y "/\" X` = Bottom L by A2,Def4;
A5: X "/\" Y` = Bottom L by A3,Def4;
(X "\/" Y) "/\" (X "\/" X`) = X "\/" Bottom L by A4,LATTICES:31;
then (X "\/" Y) "/\" Top L = X "\/" Bottom L by LATTICES:48;
then (X "\/" Y) "/\" Top L = X by LATTICES:39;
then Y "\/" X = X by LATTICES:43;
then (Y "/\" X)` [= Y` by LATTICES:def 9;
then A6: X "\/" Y [= Y` by A1,LATTICES:50;
Y [= X "\/" Y by LATTICES:22;
then A7: Y [= Y` by A6,LATTICES:25;
(X` "\/" X) "/\" (X` "\/" Y`) = X` "\/" Bottom L by A5,LATTICES:31;
then Top L "/\" (X` "\/" Y`) = X` "\/" Bottom L by LATTICES:48;
then Top L "/\" (X` "\/" Y`) = X` by LATTICES:39;
then Y` "\/" X` = X` by LATTICES:43;
then (Y` "/\" X`)` [= Y`` by LATTICES:def 9;
then Y`` "\/" X`` [= Y`` by LATTICES:50;
then Y`` "\/" X`` [= Y by LATTICES:49;
then Y`` "\/" X [= Y by LATTICES:49;
then A8: X` "\/" Y` [= Y by A1,LATTICES:49;
Y` [= X` "\/" Y` by LATTICES:22;
then A9: Y` [= Y by A8,LATTICES:25;
(Y` "\/" Y) "/\" (Y` "\/" X`) = Y` "\/" Bottom L by A4,LATTICES:31;
then Top L "/\" (Y` "\/" X`) = Y` "\/" Bottom L by LATTICES:48;
then Top L "/\" (Y` "\/" X`) = Y` by LATTICES:39;
then X` "\/" Y` = Y` by LATTICES:43;
then (X` "/\" Y`)` [= X`` by LATTICES:def 9;
then (X` "/\" Y`)` [= X by LATTICES:49;
then X`` "\/" Y`` [= X by LATTICES:50;
then X "\/" Y`` [= X by LATTICES:49;
then A10: X` "\/" Y` [= X by A1,LATTICES:49;
X` [= X` "\/" Y` by LATTICES:22;
then A11: X` [= X by A10,LATTICES:25;
(Y "\/" X) "/\" (Y "\/" Y`) = Y "\/" Bottom L by A5,LATTICES:31;
then (Y "\/" X) "/\" Top L = Y "\/" Bottom L by LATTICES:48
.= Y by LATTICES:39;
then X "\/" Y = Y by LATTICES:43;
then (X "/\" Y)` [= X` by LATTICES:def 9;
then A12: X "\/" Y [= X` by A1,LATTICES:50;
X [= X "\/" Y by LATTICES:22;
then X [= X` by A12,LATTICES:25;
hence thesis by A7,A9,A11,Def3;
end;

theorem
X \+\ Bottom L = X
proof
thus X \+\ Bottom L = (X \ Bottom L) "\/" (Bottom L \ X) by Def2
.= X "\/" (Bottom L \ X) by Th67
.= X "\/" Bottom L by Th29
.= X by LATTICES:39;
end;

theorem
X \+\ X = Bottom L
proof
thus X \+\ X = (X \ X) "\/" (X \ X) by Def2
.= X \ X by LATTICES:17
.= Bottom L by Th66;
end;

theorem
X "/\" Y misses X \+\ Y
proof
X "/\" Y misses X \ Y & X "/\" Y misses Y \ X by Th70;
then X "/\" Y misses (X \ Y) "\/" (Y \ X) by Th69;
hence thesis by Def2;
end;

theorem
X "\/" Y = X \+\ (Y \ X)
proof
X "\/" Y = (X "\/" Y) "/\" Top L by LATTICES:43
.= (X "\/" Y) "/\" (X "\/" X`) by LATTICES:48
.= X "\/" (Y "/\" X`) by LATTICES:31
.= ((Y` "/\" X) "\/" X) "\/" (Y "/\" X`) by LATTICES:def 8
.= ((X "/\" Y`) "\/" (X "/\" X)) "\/" (Y "/\" X`) by LATTICES:18
.= ((X "/\" Y`) "\/" (X "/\" X)) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:18
.= ((X "/\" Y`) "\/" (X "/\" X``)) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:49
.= (X "/\" (Y` "\/" X``)) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:def 11
.= (X "/\" (Y "/\" X`)`) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:50
.= (X "/\" (Y "/\" X`)`) "\/" ((Y "/\" X`) "/\" X`) by LATTICES:def 7
.= (X \ (Y "/\" X`)) "\/" ((Y "/\" X`) "/\" X`) by Def1
.= (X \ (Y "/\" X`)) "\/" ((Y "/\" X`) \ X) by Def1
.= X \+\ (Y "/\" X`) by Def2
.= X \+\ (Y \ X) by Def1;
hence thesis;
end;

theorem
X \+\ (X "/\" Y) = X \ Y
proof
X \+\ (X "/\" Y) = (X \ (X "/\" Y)) "\/" ((X "/\" Y) \ X) by Def2
.= (X "/\" (X "/\" Y)`) "\/" ((X "/\" Y) \ X) by Def1
.= (X "/\" (X "/\" Y)`) "\/" ((X "/\" Y) "/\" X`) by Def1
.= (X "/\" (X "/\" Y)`) "\/" (Y "/\" (X "/\" X`)) by LATTICES:def 7
.= (X "/\" (X "/\" Y)`) "\/" (Y "/\" Bottom L) by LATTICES:47
.= (X "/\" (X "/\" Y)`) "\/" Bottom L by LATTICES:40
.= X "/\" (X "/\" Y)` by LATTICES:39
.= X "/\" (X` "\/" Y`) by LATTICES:50
.= (X "/\" X`) "\/" (X "/\" Y`) by LATTICES:def 11
.= Bottom L "\/" (X "/\" Y`) by LATTICES:47
.= X "/\" Y` by LATTICES:39;
hence thesis by Def1;
end;

theorem
X "\/" Y = (X \+\ Y) "\/" (X "/\" Y)
proof
thus X "\/" Y = (Y \ X) "\/" X by Th52
.= (Y \ X) "\/" ((X \ Y) "\/" (X "/\" Y)) by Th53
.= ((Y \ X) "\/" (X \ Y)) "\/" (X "/\" Y) by LATTICES:def 5
.= (X \+\ Y) "\/" (X "/\" Y) by Def2;
end;

Lm2: (X "\/" Y) \ (X \+\ Y) = X "/\" Y
proof
set XY = X "\/" Y;
thus XY \ (X \+\ Y) = (X "\/" Y) \ ((X \ Y) "\/" (Y \ X)) by Def2
.= XY \ ((X "/\" Y`) "\/" (Y \ X)) by Def1
.= XY \ ((X "/\" Y`) "\/" (Y "/\" X`)) by Def1
.= XY "/\" ((X "/\" Y`) "\/" (Y "/\" X`))` by Def1
.= XY "/\" ((X "/\" Y`)` "/\" (Y "/\" X`)`) by LATTICES:51
.= XY "/\" ((X "/\" Y`)` "/\" (Y` "\/" X``)) by LATTICES:50
.= XY "/\" ((X` "\/" Y``) "/\" (Y` "\/" X``)) by LATTICES:50
.= XY "/\" ((X` "\/" Y``) "/\" (Y` "\/" X)) by LATTICES:49
.= XY "/\" ((X` "\/" Y) "/\" (Y` "\/" X)) by LATTICES:49
.= (XY "/\" (X` "\/" Y)) "/\" (Y` "\/" X) by LATTICES:def 7
.= ( (XY "/\" X`) "\/" ((X "\/" Y) "/\" Y) ) "/\" (Y` "\/"
X) by LATTICES:def 11
.= ( (XY "/\" X`) "\/" Y ) "/\" (Y` "\/" X) by LATTICES:def 9
.= ( ((X "/\" X`) "\/" (Y "/\" X`)) "\/" Y ) "/\" (Y` "\/"
X) by LATTICES:def 11
.= ( (Bottom L "\/" (Y "/\" X`)) "\/" Y ) "/\" (Y` "\/" X) by LATTICES:47
.= ( (Y "/\" X`) "\/" Y ) "/\" (Y` "\/" X) by LATTICES:39
.= Y "/\" (Y` "\/" X) by LATTICES:def 8
.= (Y "/\" Y`) "\/" (Y "/\" X) by LATTICES:def 11
.= Bottom L "\/" (Y "/\" X) by LATTICES:47
.= X "/\" Y by LATTICES:39;
end;

theorem
(X \+\ Y) \+\ (X "/\" Y) = X "\/" Y
proof
set XY = X \+\ Y, A = Y "/\" X`;
XY \+\ (X "/\" Y) = (XY \ (X "/\" Y)) "\/" ((X "/\" Y) \ XY) by Def2
.= (XY "/\" (X "/\" Y)`) "\/" ((X "/\" Y) \ XY) by Def1
.= (XY "/\" (X "/\" Y)`) "\/" ((X "/\" Y) "/\" XY`) by Def1
.= (XY "/\" (X` "\/" Y`)) "\/" ((X "/\" Y) "/\" XY`) by LATTICES:50
.= ( (XY "/\" X`) "\/" (XY "/\" Y`) ) "\/" ((X "/\" Y) "/\"
XY`) by LATTICES:def 11
.= ( (XY \ X) "\/" (XY "/\" Y`) ) "\/" ((X "/\" Y) "/\" XY`) by Def1
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" XY`) by Def1
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X \ Y) "\/" (Y \ X))` ) by Def2
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X "/\" Y`) "\/" (Y \ X))` ) by Def1
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X "/\" Y`) "\/" A)` ) by Def1
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X "/\" Y`)` "/\" A`) ) by LATTICES:51
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X` "\/" Y``) "/\" A`) ) by LATTICES:50
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X` "\/" Y``) "/\" (Y` "\/" X``)) ) by LATTICES:50
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X` "\/" Y) "/\" (Y` "\/" X``)) ) by LATTICES:49
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y)
"/\" ((X` "\/" Y) "/\" (Y` "\/" X)) ) by LATTICES:49
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
(((X "/\" Y) "/\" (X` "\/" Y)) "/\" (Y` "\/"
X)) by LATTICES:def 7
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
( (((X "/\" Y) "/\" X`) "\/" ((X "/\" Y) "/\" Y)) "/\" (Y` "\/"
X)) by LATTICES:def 11
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
( ((Y "/\" (X "/\" X`)) "\/" ((X "/\" Y) "/\" Y)) "/\" (Y` "\/"
X)) by LATTICES:def 7
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
( ((Y "/\" Bottom L) "\/" ((X "/\" Y) "/\" Y)) "/\" (Y` "\/"
X)) by LATTICES:47
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
( (Bottom L "\/" ((X "/\" Y) "/\" Y)) "/\" (Y` "\/" X)) by LATTICES:40
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
(((X "/\" Y) "/\" Y) "/\" (Y` "\/" X)) by LATTICES:39
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
((X "/\" (Y "/\" Y)) "/\" (Y` "\/" X)) by LATTICES:def 7
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" (Y` "\/"
X)) by LATTICES:18
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
(((X "/\" Y) "/\" Y`) "\/" ((X "/\" Y) "/\" X)) by LATTICES:def 11
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
((X "/\" (Y "/\" Y`)) "\/" ((X "/\" Y) "/\" X)) by LATTICES:def 7
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
((X "/\" Bottom L) "\/" ((X "/\" Y) "/\" X)) by LATTICES:47
.= ( (XY \ X) "\/" (XY \ Y) ) "\/"
( Bottom L "\/" ((X "/\" Y) "/\" X)) by LATTICES:40
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" X) by LATTICES:39
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" (Y "/\" (X "/\" X)) by LATTICES:def 7
.= ( (XY \ X) "\/" (XY \ Y) ) "\/" (Y "/\" X) by LATTICES:18
.= ( (XY "/\" X`) "\/" (XY \ Y) ) "\/" (Y "/\" X) by Def1
.= ( (XY "/\" X`) "\/" (XY "/\" Y`) ) "\/" (Y "/\" X) by Def1
.= ( (((X \ Y) "\/" (Y \ X)) "/\" X`) "\/" (XY "/\" Y`) ) "\/" (Y "/\"
X) by Def2
.= ( (((X \ Y) "\/" (Y \ X)) "/\" X`) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by Def2
.= ( (((X "/\" Y`) "\/" (Y \ X)) "/\" X`) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by Def1
.= ( (((X "/\" Y`) "\/" A) "/\" X`) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by Def1
.= ( (((X "/\" Y`) "/\" X`) "\/" (A "/\" X`)) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:def 11
.= ( ((Y` "/\" (X "/\" X`)) "\/" (A "/\" X`)) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:def 7
.= ( ((Y` "/\" (X "/\" X`)) "\/" (Y "/\" (X` "/\" X`))) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:def 7
.= ( ((Y` "/\" Bottom L) "\/" (Y "/\" (X` "/\" X`))) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:47
.= ( ((Y` "/\" Bottom L) "\/" A) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:18
.= ( (Bottom L "\/" A) "\/"
(((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:40
.= ( A "\/" (((X \ Y) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\"
X) by LATTICES:39
.= ( A "\/" (((X "/\" Y`) "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by Def1
.= ( A "\/" (((X "/\" Y`) "\/" A) "/\" Y`) ) "\/" (Y "/\" X) by Def1
.= ( A "\/" (((X "/\" Y`) "/\" Y`) "\/" (A "/\" Y`)) )
"\/" (Y "/\" X) by LATTICES:def 11
.= ( A "\/" ( (X "/\" (Y` "/\" Y`)) "\/" (A "/\" Y`)) )
"\/" (Y "/\" X) by LATTICES:def 7
.= ( A "\/" ((X "/\" Y`) "\/" (A "/\" Y`)) ) "\/" (Y "/\"
X) by LATTICES:18
.= ( A "\/" ( (X "/\" Y`) "\/" (X` "/\" (Y "/\" Y`))) )
"\/" (Y "/\" X) by LATTICES:def 7
.= ( A "\/" ((X "/\" Y`) "\/" (X` "/\" Bottom L) )) "\/" (Y "/\"
X) by LATTICES:47
.= ( A "\/" ((X "/\" Y`) "\/" Bottom L )) "\/" (Y "/\" X) by LATTICES:40
.= ( A "\/" (X "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:39
.= A "\/" ( (X "/\" Y`) "\/" (Y "/\" X) ) by LATTICES:def 5
.= A "\/" (X "/\" (Y` "\/" Y) ) by LATTICES:def 11
.= A "\/" (X "/\" Top L) by LATTICES:48
.= A "\/" X by LATTICES:43
.= (Y "\/" X) "/\" (X` "\/" X ) by LATTICES:31
.= (Y "\/" X) "/\" Top L by LATTICES:48
.= Y "\/" X by LATTICES:43;
hence thesis;
end;

theorem
(X \+\ Y) \+\ (X "\/" Y) = X "/\" Y
proof
(X \+\ Y) \+\ (X "\/" Y) = ((X \+\ Y) \ (X "\/" Y)) "\/" ((X "\/" Y) \ (X
\+\
Y)) by Def2
.= ((X \+\ Y) "/\" (X "\/" Y)`) "\/" ((X "\/" Y) \ (X \+\ Y)) by Def1
.= ((X \+\ Y) "/\" (X` "/\" Y`)) "\/" ((X "\/"
Y) \ (X \+\ Y)) by LATTICES:51
.= ( ((X \ Y) "\/" (Y \ X)) "/\" (X` "/\" Y`)) "\/" ((X "\/"
Y) \ (X \+\ Y)) by Def2
.= ( ((X "/\" Y`) "\/" (Y \ X)) "/\" (X` "/\" Y`)) "\/" ((X "\/"
Y) \ (X \+\ Y)) by Def1
.= ( ((X "/\" Y`) "\/" (Y "/\" X`)) "/\" (X` "/\" Y`))
"\/" ((X "\/"
Y) \ (X \+\ Y)) by Def1
.= ( ((X "/\" Y`) "/\" (X` "/\" Y`)) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 11
.= ( (X "/\" (Y` "/\" (Y` "/\" X`))) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
.= ( (X "/\" ((Y` "/\" Y`) "/\" X`)) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
.= ( (X "/\" (Y` "/\" X`)) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:18
.= ( ((X "/\" X`) "/\" Y`) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
.= ( (Bottom L "/\" Y`) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:47
.= ( Bottom L "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) )
"\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:40
.= ((Y "/\" X`) "/\" (X` "/\" Y`)) "\/" ((X "\/"
Y) \ (X \+\ Y)) by LATTICES:39
.= (Y "/\" (X` "/\" (X` "/\" Y`))) "\/" ((X "\/"
Y) \ (X \+\ Y)) by LATTICES:def 7
.= (Y "/\" ((X` "/\" X`) "/\" Y`)) "\/" ((X "\/"
Y) \ (X \+\ Y)) by LATTICES:def 7
.= (Y "/\" (X` "/\" Y`)) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:18
.= ((Y "/\" Y`) "/\" X`) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
.= (Bottom L "/\" X`) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:47
.= Bottom L "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:40
.= (X "\/" Y) \ (X \+\ Y) by LATTICES:39
.= Y "/\" X by Lm2;
hence thesis;
end;

theorem
Th84:  X \+\ Y = (X "\/" Y) \ (X "/\" Y)
proof
thus X \+\ Y = (X \ Y) "\/" (Y \ X) by Def2
.= (X \ X "/\" Y) "\/" (Y \ X) by Th50
.= (X \ X "/\" Y) "\/" (Y \ X "/\" Y) by Th50
.= (X "\/" Y) \ (X "/\" Y) by Th24;
end;

theorem
(X \+\ Y) \ Z = (X \ (Y "\/" Z)) "\/" (Y \ (X "\/" Z))
proof
thus (X \+\ Y) \ Z = ((X \ Y) "\/" (Y \ X)) \ Z by Def2
.= (X \ Y \ Z) "\/" (Y \ X \ Z) by Th24
.= (X \ (Y "\/" Z)) "\/" (Y \ X \ Z) by Th62
.= (X \ (Y "\/" Z)) "\/" (Y \ (X "\/" Z)) by Th62;
end;

theorem
X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" (X "/\" Y "/\" Z)
proof
X \ (Y \+\ Z) = X \ ((Y "\/" Z) \ (Y "/\" Z)) by Th84
.= (X \ (Y "\/" Z)) "\/" (X "/\" (Y "/\" Z)) by Th54
.= (X \ (Y "\/" Z)) "\/" (X "/\" Y "/\" Z) by LATTICES:def 7;
hence thesis;
end;

theorem
(X \+\ Y) \+\ Z = X \+\ (Y \+\ Z)
proof
set S1 = X \ (Y "\/" Z), S2 = Y \ (X "\/" Z),
S3 = Z \ (X "\/" Y), S4 = X "/\" Y "/\" Z;
thus (X \+\ Y) \+\ Z = ((X \ Y) "\/" (Y \ X)) \+\ Z by Def2
.= (((X \ Y) "\/" (Y \ X)) \ Z) "\/" (Z \ ((X \ Y) "\/" (Y \ X))) by Def2
.= (((X \ Y) \ Z) "\/" ((Y \ X) \ Z)) "\/" (Z \ ((X \ Y) "\/" (Y \ X)))
by Th24
.= ( S1 "\/" ((Y \ X) \ Z)) "\/" (Z \ ((X \ Y) "\/" (Y \ X))) by Th62
.= ( S1 "\/" S2) "\/" (Z \ ((X \ Y) "\/" (Y \ X))) by Th62
.= ( S1 "\/" S2) "\/" (Z \ ((X "\/" Y) \ (X "/\" Y))) by Th61
.= ( S1 "\/" S2) "\/" (S3 "\/" (X "/\" Y "/\" Z )) by Th54
.= (S1 "\/" S2 "\/" S4) "\/" S3 by LATTICES:def 5
.= (S1 "\/" S4 "\/" S2) "\/" S3 by LATTICES:def 5
.= (S1 "\/" S4) "\/" (S2 "\/" S3) by LATTICES:def 5
.= (S1 "\/" (X "/\" (Y "/\" Z))) "\/" (S2 "\/" S3) by LATTICES:def 7
.= (X \ ((Y "\/" Z) \ (Y "/\" Z))) "\/" (S2 "\/" S3) by Th54
.= (X \ ((Y \ Z) "\/" (Z \ Y))) "\/" (S2 "\/" S3) by Th61
.= (X \ ((Y \ Z) "\/" (Z \ Y))) "\/" (S2 "\/" (Z \ Y \ X)) by Th62
.= (X \ ((Y \ Z) "\/" (Z \ Y))) "\/" ((Y \ Z \ X) "\/"
(Z \ Y \ X)) by Th62
.= (X \ ((Y \ Z) "\/" (Z \ Y))) "\/" (((Y \ Z) "\/" (Z \ Y)) \ X) by Th24
.= X \+\ ((Y \ Z) "\/" (Z \ Y)) by Def2
.= X \+\ (Y \+\ Z) by Def2;
end;

theorem
(X \+\ Y)` = (X "/\" Y) "\/" (X` "/\" Y`)
proof
thus (X \+\ Y)` = ((X \ Y) "\/" (Y \ X))` by Def2
.=(X \ Y)` "/\" (Y \ X)` by LATTICES:51
.=(X "/\" Y`)` "/\" (Y \ X)` by Def1
.=(X "/\" Y`)` "/\" (Y "/\" X`)` by Def1
.=(X` "\/" Y``) "/\" (Y "/\" X`)` by LATTICES:50
.=(X` "\/" Y``) "/\" (Y` "\/" X``) by LATTICES:50
.=(X` "\/" Y) "/\" (Y` "\/" X``) by LATTICES:49
.=(X` "\/" Y) "/\" (Y` "\/" X) by LATTICES:49
.=(X` "/\" (Y` "\/" X)) "\/" (Y "/\" (Y` "\/" X)) by LATTICES:def 11
.=((X` "/\" Y`) "\/" (X` "/\" X)) "\/" (Y "/\" (Y` "\/"
X)) by LATTICES:def 11
.=((X` "/\" Y`) "\/" (X` "/\" X)) "\/" ((Y "/\" Y`) "\/" (Y "/\"
X)) by LATTICES:def 11
.=((X` "/\" Y`) "\/" Bottom L) "\/" ((Y "/\" Y`) "\/" (Y "/\"
X)) by LATTICES:47
.=((X` "/\" Y`) "\/" Bottom L) "\/" (Bottom
L "\/" (Y "/\" X)) by LATTICES:47
.=(X` "/\" Y`) "\/" (Bottom L "\/" (Y "/\" X)) by LATTICES:39
.=(X "/\" Y) "\/" (X` "/\" Y`) by LATTICES:39;
end;
```