### The Mizar article:

### 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;

Back to top