Introduction to Lattice Theory

by
Stanislaw Zukowski

Copyright (c) 1989 Association of Mizar Users

MML identifier: LATTICES
[ MML identifier index ]

environ

vocabulary BINOP_1, BOOLE, FINSUB_1, FUNCT_1, SUBSET_1, LATTICES;
notation XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, BINOP_1, FINSUB_1;
constructors STRUCT_0, BINOP_1, FINSUB_1, XBOOLE_0;
clusters FINSUB_1, STRUCT_0, SUBSET_1, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;
theorems TARSKI, ZFMISC_1, STRUCT_0, FINSUB_1;

begin

definition
struct(1-sorted) /\-SemiLattStr
(# carrier -> set, L_meet -> BinOp of the carrier #);
end;

definition
struct(1-sorted) \/-SemiLattStr
(# carrier -> set, L_join -> BinOp of the carrier #);
end;

definition
struct(/\-SemiLattStr,\/-SemiLattStr) LattStr
(# carrier -> set, L_join, L_meet -> BinOp of the carrier #);
end;

set DD=bool {};

DD is preBoolean
proof
now let X,Y be set;
assume X in DD & Y in DD;
then reconsider X'=X,Y'=Y as Element of DD;
X' \/ Y' in DD & X' \ Y' in DD;
hence X \/ Y in DD & X \ Y in DD;
end;
hence thesis by FINSUB_1:10;
end;

then reconsider DD as non empty preBoolean set;

consider uu,nn being BinOp of DD;
set GG=LattStr(#DD,uu,nn#);
set GGj=\/-SemiLattStr(#DD,uu#);
set GGm=/\-SemiLattStr(#DD,uu#);

Lm1: GG is strict non empty by STRUCT_0:def 1;
Lm2: GGj is strict non empty by STRUCT_0:def 1;
Lm3: GGm is strict non empty by STRUCT_0:def 1;

definition
cluster strict non empty \/-SemiLattStr;
existence by Lm2;
cluster strict non empty /\-SemiLattStr;
existence by Lm3;
cluster strict non empty LattStr;
existence by Lm1;
end;

reconsider GG as strict non empty LattStr by STRUCT_0:def 1;
reconsider GGj as strict non empty \/-SemiLattStr by STRUCT_0:def 1;
reconsider GGm as strict non empty /\-SemiLattStr by STRUCT_0:def 1;

definition let G be non empty \/-SemiLattStr,
p, q be Element of G;
func p"\/"q -> Element of G equals
(the L_join of G).(p,q);
coherence;
end;

definition let G be non empty /\-SemiLattStr,
p, q be Element of G;
func p"/\"q -> Element of G equals
(the L_meet of G).(p,q);
coherence;
end;

definition let G be non empty \/-SemiLattStr,
p, q be Element of G;
pred p [= q means :Def3:
p"\/"q = q;
end;

Lm4: for x,y being Element of GG holds x = y
proof let x,y be Element of GG;
x = {} & y = {} by TARSKI:def 1,ZFMISC_1:1;
hence thesis;
end;

Lm5: for x,y being Element of GGj holds x = y
proof let x,y be Element of GGj;
x = {} & y = {} by TARSKI:def 1,ZFMISC_1:1;
hence thesis;
end;

Lm6: for x,y being Element of GGm holds x = y
proof let x,y be Element of GGm;
x = {} & y = {} by TARSKI:def 1,ZFMISC_1:1;
hence thesis;
end;

Lm7: for x,y being Element of GG holds x"\/"y = y"\/"x by Lm4;

Lm8: for x,y being Element of GGj holds x"\/"y = y"\/"x by Lm5;

Lm9: for x,y being Element of GGm holds x"/\"y = y"/\"x by Lm6;

Lm10: for x,y,z being Element of GG holds
x"\/"(y"\/"z) = (x"\/"y)"\/"z by Lm4;

Lm11: for x,y,z being Element of GGj holds
x"\/"(y"\/"z) = (x"\/"y)"\/"z by Lm5;

Lm12: for x,y,z being Element of GGm holds
x"/\"(y"/\"z) = (x"/\"y)"/\"z by Lm6;

Lm13: for x,y being Element of GG holds
(x"/\"y)"\/"y = y by Lm4;

Lm14: for x,y being Element of GG holds
x"/\"y = y"/\"x by Lm4;

Lm15: for x,y,z being Element of GG holds
x"/\"(y"/\"z) = (x"/\"y)"/\"z by Lm4;

Lm16: for x,y being Element of GG holds
x"/\"(x"\/"y) = x by Lm4;

Lm17: for x,y,z being Element of GG holds
x"/\"(y"\/"z) = (x"/\"y)"\/"(x"/\"z) by Lm4;

Lm18: for x,y,z being Element of GG holds
x [= z implies x"\/"(y"/\"z) = (x"\/"y)"/\"z by Lm4;
reconsider 0_GG={} as Element of GG by ZFMISC_1:def 1;
reconsider D={} as Element of GG by ZFMISC_1:def 1;

Lm19: for x being Element of GG holds
0_GG"/\"x = 0_GG & x"/\"0_GG = 0_GG by Lm4;
Lm20: for x being Element of GG holds
D"\/"x = D & x"\/"D = D by Lm4;

definition let IT be non empty \/-SemiLattStr;
attr IT is join-commutative means :Def4:
for a,b being Element of IT holds a"\/"b = b"\/"a;
attr IT is join-associative means :Def5:
for a,b,c being Element of IT holds a"\/"(b"\/"c) = (a"\/"b)
"\/"c;
end;

definition let IT be non empty /\-SemiLattStr;
attr IT is meet-commutative means :Def6:
for a,b being Element of IT holds a"/\"b = b"/\"a;
attr IT is meet-associative means :Def7:
for a,b,c being Element of IT holds a"/\"(b"/\"c) = (a"/\"b)
"/\"c;
end;

definition let IT be non empty LattStr;
attr IT is meet-absorbing means :Def8:
for a,b being Element of IT holds (a"/\"b)"\/"b = b;
attr IT is join-absorbing means :Def9:
for a,b being Element of IT holds a"/\"(a"\/"b)=a;
end;

definition let IT be non empty LattStr;
attr IT is Lattice-like means :Def10:
IT is join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing;
end;

definition
cluster Lattice-like ->
join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
(non empty LattStr);
coherence by Def10;
cluster join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
-> Lattice-like (non empty LattStr);
coherence by Def10;
end;

definition
cluster strict join-commutative join-associative (non empty \/-SemiLattStr);
existence
proof GGj is join-commutative join-associative by Def4,Def5,Lm8,Lm11;
hence thesis;
end;
cluster strict meet-commutative meet-associative (non empty /\-SemiLattStr);
existence
proof GGm is meet-commutative meet-associative by Def6,Def7,Lm9,Lm12;
hence thesis;
end;
cluster strict Lattice-like (non empty LattStr);
existence
proof GG is join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
by Def4,Def5,Def6,Def7,Def8,Def9,Lm7,Lm10,Lm13,Lm14,Lm15,Lm16;
then GG is Lattice-like by Def10;
hence thesis;
end;
end;

definition
mode Lattice is Lattice-like (non empty LattStr);
end;

GG is join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
by Def4,Def5,Def6,Def7,Def8,Def9,Lm7,Lm10,Lm13,Lm14,Lm15,Lm16;
then reconsider GG as Lattice by Def10;

definition let L be join-commutative (non empty \/-SemiLattStr),
a, b be Element of L;
redefine func a"\/"b;
commutativity by Def4;
end;

definition let L be meet-commutative (non empty /\-SemiLattStr),
a, b be Element of L;
redefine func a"/\"b;
commutativity by Def6;
end;

definition let IT be non empty LattStr;
attr IT is distributive means :Def11:
for a,b,c being Element of IT
holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c);
end;

Lm21: GG is distributive by Def11,Lm17;

definition let IT be non empty LattStr;
attr IT is modular means :Def12:
for a,b,c being Element of IT st a [= c
holds a"\/"(b"/\"c) = (a"\/"b)"/\"c;
end;

definition let IT be non empty /\-SemiLattStr;
attr IT is lower-bounded means :Def13:
ex c being Element of IT st
for a being Element of IT holds c"/\"a = c & a"/\"c = c;
end;

definition let IT be non empty \/-SemiLattStr;
attr IT is upper-bounded means :Def14:
ex c being Element of IT st
for a being Element of IT holds c"\/"a = c & a"\/"c = c;
end;

definition
cluster strict distributive lower-bounded upper-bounded modular Lattice;
existence
proof A1: GG is modular by Def12,Lm18;
A2: GG is lower-bounded by Def13,Lm19;
GG is upper-bounded by Def14,Lm20;
hence thesis by A1,A2,Lm21;
end;
end;

definition
mode D_Lattice is distributive Lattice;
mode M_Lattice is modular Lattice;
mode 0_Lattice is lower-bounded Lattice;
mode 1_Lattice is upper-bounded Lattice;
end;

Lm22: GG is 0_Lattice by Def13,Lm19;
Lm23: GG is 1_Lattice by Def14,Lm20;

definition let IT be non empty LattStr;
attr IT is bounded means :Def15:
IT is lower-bounded upper-bounded;
end;

definition
cluster lower-bounded upper-bounded -> bounded (non empty LattStr);
coherence by Def15;
cluster bounded -> lower-bounded upper-bounded (non empty LattStr);
coherence by Def15;
end;

definition
cluster bounded strict Lattice;
existence
proof
GG is 0_Lattice & GG is 1_Lattice by Def13,Def14,Lm19,Lm20;
then GG is bounded by Def15;
hence thesis;
end;
end;

definition
mode 01_Lattice is bounded Lattice;
end;

definition let L be non empty /\-SemiLattStr;
assume A1: L is lower-bounded;
func Bottom L -> Element of L means :Def16:
for a being Element of L holds it "/\" a = it & a "/\"
it = it;
existence by A1,Def13;
uniqueness
proof
let c1,c2 be Element of L such that
A2:  for a being Element of L holds c1"/\"a = c1 & a"/\"c1 = c1
and
A3:  for a being Element of L holds c2"/\"a = c2 & a"/\"c2 = c2;
thus c1 = c2"/\"c1 by A2
.= c2 by A3;
end;
end;

definition let L be non empty \/-SemiLattStr;
assume A1: L is upper-bounded;
func Top L -> Element of L means :Def17:
for a being Element of L holds it "\/" a = it & a "\/"
it = it;
existence by A1,Def14;
uniqueness
proof
let c1,c2 be Element of L such that
A2:  for a being Element of L holds c1"\/"a = c1 & a"\/"
c1 = c1 and
A3:  for a being Element of L holds c2"\/"a = c2 & a"\/"c2 = c2;
thus c1 = c2"\/"c1 by A2
.= c2 by A3;
end;
end;

definition let L be non empty LattStr,
a, b be Element of L;
pred a is_a_complement_of b means :Def18:
a"\/"b = Top L & b"\/"a = Top L & a"/\"b = Bottom L & b"/\"a = Bottom L;
end;

definition let IT be non empty LattStr;
attr IT is complemented means :Def19:
for b being Element of IT
ex a being Element of IT st a is_a_complement_of b;
end;

definition
cluster bounded complemented strict Lattice;
existence
proof
take GG;
thus GG is bounded by Def15,Lm22,Lm23;
thus GG is complemented
proof let b be Element of GG;
consider a being Element of GG;
take a;
thus a"\/"b = Top GG & b"\/"a = Top GG by Lm4;
thus a"/\"b = Bottom GG & b"/\"a = Bottom GG by Lm4;
end;
thus thesis;
end;
end;

definition
mode C_Lattice is complemented 01_Lattice;
end;

reconsider GG as 01_Lattice by Def15,Lm22,Lm23;

Lm24: GG is complemented
proof
let b be Element of GG;
take b;
thus b"\/"b = Top GG & b"\/"b = Top GG by Lm4;
thus b"/\"b = Bottom GG & b"/\"b = Bottom GG by Lm4;
end;

definition let IT be non empty LattStr;
attr IT is Boolean means :Def20:
IT is bounded complemented distributive;
end;

definition
cluster Boolean -> bounded complemented distributive (non empty LattStr);
coherence by Def20;
cluster bounded complemented distributive -> Boolean (non empty LattStr);
coherence by Def20;
end;

definition
cluster Boolean strict Lattice;
existence
proof reconsider GG as C_Lattice by Lm24;
GG is Boolean by Def20,Lm21;
hence thesis;
end;
end;

definition
mode B_Lattice is Boolean Lattice;
end;

reserve L for meet-absorbing join-absorbing meet-commutative
(non empty LattStr);
reserve a for Element of L;

canceled 16;

theorem Th17:
a"\/"a = a
proof
thus a"\/"a = ( a "/\" ( a"\/"a ) ) "\/" a by Def9
.= a by Def8;
end;

theorem
a"/\"a = a
proof
a"/\"( a"\/"a ) = a by Def9;
hence thesis by Th17;
end;

reserve L for Lattice;
reserve a, b, c for Element of L;

theorem Th19:
(for a,b,c holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c))
iff
(for a,b,c holds a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c))
proof
hereby
assume A1:for a,b,c holds a"/\"(b"\/"c)=(a"/\"b)"\/"(a"/\"c);
let a,b,c;
thus a"\/"(b"/\"c)=(a"\/"(c"/\"a))"\/"(c"/\"b) by Def8
.=a"\/"((c"/\"a)"\/"(c"/\"b)) by Def5
.=a"\/"((a"\/"b)"/\"c) by A1
.=((a"\/"b)"/\"a)"\/"((a"\/"b)"/\"c) by Def9
.=(a"\/"b)"/\"(a"\/"c) by A1;
end;
assume A2:for a,b,c holds a"\/"(b"/\"c)=(a"\/"b)"/\"(a"\/"c);
let a,b,c;
thus a"/\"(b"\/"c)=(a"/\"(c"\/"a))"/\"(c"\/"b) by Def9
.=a"/\"((c"\/"a)"/\"(c"\/"b)) by Def7
.=a"/\"((a"/\"b)"\/"c) by A2
.=((a"/\"b)"\/"a)"/\"((a"/\"b)"\/"c) by Def8
.=(a"/\"b)"\/"(a"/\"c) by A2;
end;

canceled;

theorem Th21:
for L being meet-absorbing join-absorbing (non empty LattStr),
a, b being Element of L holds
a [= b iff a"/\"b = a
proof
let L be meet-absorbing join-absorbing (non empty LattStr),
a, b be Element of L;
a [= b iff a"\/"b = b by Def3;
hence thesis by Def8,Def9;
end;

theorem Th22:
for L being meet-absorbing join-absorbing join-associative meet-commutative
(non empty LattStr),
a, b being Element of L holds
a [= a"\/"b
proof
let L be meet-absorbing join-absorbing join-associative meet-commutative
(non empty LattStr),
a, b be Element of L;
thus a"\/"( a"\/"b) = (a"\/"a)"\/"b by Def5
.= a"\/"b by Th17;
end;

theorem Th23:
for L being meet-absorbing meet-commutative (non empty LattStr),
a, b being Element of L holds
a"/\"b [= a
proof
let L be meet-absorbing meet-commutative (non empty LattStr),
a, b be Element of L;
thus ( a"/\"b )"\/"a = a by Def8;
end;

definition
let L be meet-absorbing join-absorbing meet-commutative (non empty LattStr),
a, b be Element of L;
redefine pred a [= b;
reflexivity
proof let a be Element of L;
thus a"\/"a = a by Th17;
end;
end;

canceled;

theorem
for L being join-associative (non empty \/-SemiLattStr),
a, b, c being Element of L holds
a [= b & b [= c implies a [= c
proof
let L be join-associative (non empty \/-SemiLattStr),
a, b, c be Element of L;
assume a"\/"b = b & b"\/"c = c;
hence a"\/"c = c by Def5;
end;

theorem Th26:
for L being join-commutative (non empty \/-SemiLattStr),
a, b being Element of L holds
a [= b & b [= a implies a=b
proof
let L be join-commutative (non empty \/-SemiLattStr),
a, b be Element of L;
assume a"\/"b = b & b"\/"a =a;
hence thesis;
end;

theorem Th27:
for L being meet-absorbing join-absorbing meet-associative (non empty LattStr),
a, b, c being Element of L holds
a [= b implies a"/\"c [= b"/\"c
proof
let L be meet-absorbing join-absorbing meet-associative (non empty LattStr),
a, b, c be Element of L;
assume a [= b;
hence (a"/\"c)"\/"(b"/\"c) = ((a"/\"b)"/\"c)"\/"(b"/\"c) by Th21
.= (a"/\"(b"/\"c))"\/"(b"/\"c) by Def7
.= b"/\"c by Def8;
end;

canceled;

theorem
for L being Lattice holds
(for a,b,c being Element of L holds
(a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a))
implies L is distributive
proof
let L be Lattice;
assume
A1:   for a,b,c being Element of L holds
(a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a);
A2:    now let a,b,c be Element of L;
assume A3: c [= a;
thus a"/\"(b"\/"c) = (b"\/"c)"/\"(a"/\"(a"\/"b)) by Def9
.= (a"\/"b)"/\"((b"\/"c)"/\"a) by Def7
.= (a"\/"b)"/\"((b"\/"c)"/\"(c"\/"a)) by A3,Def3
.= (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) by Def7
.= (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) by A1
.= (a"/\"b)"\/"((b"/\"c)"\/"(c"/\"a)) by Def5
.= (a"/\"b)"\/"((b"/\"c)"\/"c) by A3,Th21
.= (a"/\"b)"\/"c by Def8;
end;
let a,b,c be Element of L;
A4:   (a"/\"b)"\/"(c"/\"a) [= a
proof
thus ((a"/\"b)"\/"(c"/\"a))"\/"a = (a"/\"b)"\/"((c"/\"a)"\/"a) by Def5
.= (a"/\"b)"\/"a by Def8
.= a by Def8;
end;
thus a"/\"(b"\/"c)
= (a"/\"(c"\/"a))"/\"(b"\/"c) by Def9
.= a"/\"((c"\/"a)"/\"(b"\/"c)) by Def7
.= (a"/\"(a"\/"b))"/\"((c"\/"a)"/\"(b"\/"c)) by Def9
.= a"/\"((a"\/"b)"/\"((b"\/"c)"/\"(c"\/"a))) by Def7
.= a"/\"((a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a)) by Def7
.= a"/\"((b"/\"c)"\/"(a"/\"b)"\/"(c"/\"a)) by A1
.= a"/\"((b"/\"c)"\/"((a"/\"b)"\/"(c"/\"a))) by Def5
.= (a"/\"(b"/\"c))"\/"((a"/\"b)"\/"(c"/\"a)) by A2,A4
.= (a"/\"b"/\"c)"\/"((a"/\"b)"\/"(c"/\"a)) by Def7
.= ((a"/\"b"/\"c)"\/"(a"/\"b))"\/"(c"/\"a) by Def5
.= (a"/\"b)"\/"(a"/\"c) by Def8;
end;

reserve L for D_Lattice;
reserve a, b, c for Element of L;

canceled;

theorem Th31:
a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c)
proof
for a,b,c holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c) by Def11;
hence thesis by Th19;
end;

theorem Th32:
c"/\"a = c"/\"b & c"\/"a = c"\/"b implies a=b
proof
assume that
A1:           c"/\"a = c"/\"b and
A2:           c"\/"a = c"\/"b;
thus a = a"/\"( c"\/"a ) by Def9
.= ( b"/\"c )"\/"( b"/\"a ) by A1,A2,Def11
.= b"/\"( c"\/"a ) by Def11
.= b by A2,Def9;
end;

canceled;

theorem
(a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) = (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a)
proof
thus (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a)
= (((a"\/"b)"/\"(b"\/"c))"/\"c)"\/"(((a"\/"b)"/\"(b"\/"c))"/\"a) by Def11
.= ((a"\/"b)"/\"((b"\/"c)"/\"c))"\/"(((a"\/"b)"/\"(b"\/"c))"/\"a) by Def7
.= ((a"\/"b)"/\"c)"\/"(a"/\"((a"\/"b)"/\"(b"\/"c))) by Def9
.= ((a"\/"b)"/\"c)"\/"((a"/\"(a"\/"b))"/\"(b"\/"c)) by Def7
.= (c"/\"(a"\/"b))"\/"(a"/\"(b"\/"c)) by Def9
.= ((c"/\"a)"\/"(c"/\"b))"\/"(a"/\"(b"\/"c)) by Def11
.= ((a"/\"b)"\/"(c"/\"a))"\/"((c"/\"a)"\/"(b"/\"c)) by Def11
.= (a"/\"b)"\/"((c"/\"a)"\/"((c"/\"a)"\/"(b"/\"c))) by Def5
.= (a"/\"b)"\/"(((c"/\"a)"\/"(c"/\"a))"\/"(b"/\"c)) by Def5
.= (a"/\"b)"\/"((b"/\"c)"\/"(c"/\"a)) by Th17
.= (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) by Def5;
end;

definition
cluster distributive -> modular Lattice;
coherence
proof let L be Lattice;
assume A1: L is distributive;
let a,b,c be Element of L;
assume a"\/"c = c;
hence a"\/"(b"/\"c) = (a"\/"b)"/\"c by A1,Th31;
end;
end;

reserve L for 0_Lattice;
reserve a for Element of L;

canceled 4;

theorem Th39:
Bottom L"\/"a = a
proof
thus Bottom L"\/"a = ( Bottom L"/\"a )"\/"a by Def16
.= a by Def8;
end;

theorem
Bottom L"/\"a = Bottom L by Def16;

theorem Th41:
Bottom L [= a
proof
Bottom L [= Bottom L"\/"a by Th22;
hence thesis by Th39;
end;

reserve L for 1_Lattice;
reserve a for Element of L;

canceled;

theorem Th43:
Top L"/\"a = a
proof
thus Top L"/\"a = ( Top L"\/"a )"/\"a by Def17
.= a by Def9;
end;

theorem
Top L"\/"a = Top L by Def17;

theorem
a [= Top L
proof
Top L"/\"a [= Top L by Th23;
hence thesis by Th43;
end;

definition let L be non empty LattStr,
x be Element of L;
assume A1: L is complemented D_Lattice;
func x` -> Element of L means :Def21:
it is_a_complement_of x;
existence by A1,Def19;
uniqueness
proof
let a,b be Element of L such that
A2:           a is_a_complement_of x and
A3:           b is_a_complement_of x;
A4:  a"\/"x = Top L & x"\/"a = Top L & a"/\"x = Bottom L & x"/\"a = Bottom
L by A2,Def18;
b"\/"x = Top L & b"/\"x = Bottom L & x"\/"b = Top L & x"/\"b = Bottom
L by A3,Def18;
hence thesis by A1,A4,Th32;
end;
end;

reserve L for B_Lattice;
reserve a, b for Element of L;

canceled;

theorem Th47:
a`"/\"a = Bottom L
proof a` is_a_complement_of a by Def21;
hence thesis by Def18;
end;

theorem Th48:
a`"\/"a = Top L
proof a` is_a_complement_of a by Def21;
hence thesis by Def18;
end;

theorem Th49:
a`` = a
proof
a`` is_a_complement_of a` by Def21;
then A1:   a``"\/"a` = Top L & a``"/\"a` = Bottom L by Def18;
a` is_a_complement_of a by Def21;
then a"\/"a` =Top L & a "/\"a`= Bottom L by Def18;
hence a``= a by A1,Th32;
end;

theorem Th50:
( a"/\"b )` = a`"\/" b`
proof
A1:    (a`"\/"b`)"\/"(a"/\"b) = a`"\/"(b`"\/"(a"/\"b)) by Def5
.= a`"\/"((b`"\/"a)"/\"(b`"\/"b)) by Th31
.= a`"\/"((b`"\/"a)"/\"Top L) by Th48
.= (b`"\/"a)"\/"a` by Th43
.= b`"\/"(a"\/"a`) by Def5
.= b`"\/"Top L by Th48
.= Top L by Def17;
(a`"\/"b`)"/\"(a"/\"b) = ((a`"\/"b`)"/\"a)"/\"b by Def7
.= ((a`"/\"a)"\/"(b`"/\"a))"/\"b by Def11
.= (Bottom L"\/"(b`"/\"a))"/\"b by Th47
.= b"/\"(b`"/\"a) by Th39
.= (b"/\"b`)"/\"a by Def7
.= Bottom L"/\"a by Th47
.= Bottom L by Def16;
then a`"\/"b` is_a_complement_of a"/\"b by A1,Def18;
hence thesis by Def21;
end;

theorem
( a"\/"b )` = a`"/\" b`
proof
thus (a"\/"b)` = (a``"\/"b)` by Th49
.= (a``"\/"b``)` by Th49
.= (a`"/\"b`)`` by Th50
.= a`"/\"b` by Th49;
end;

theorem Th52:
b"/\"a = Bottom L iff b [= a`
proof
thus b"/\"a = Bottom L implies b [= a`
proof
assume A1: b"/\"a = Bottom L;
b = b"/\"Top L by Th43
.= b"/\"(a"\/"a`) by Th48
.= Bottom L"\/"(b"/\"a`) by A1,Def11
.= b"/\"a` by Th39;
then b"\/"a` = a` by Def8;
hence b [= a` by Def3;
end;
thus thesis
proof
assume b [= a`;
then b"/\"a [= a`"/\"a by Th27;
then A2:        b"/\"a [= Bottom L by Th47;
Bottom L [= b"/\"a by Th41;
hence b"/\"a = Bottom L by A2,Th26;
end;
end;

theorem
a [= b implies b` [= a`
proof
assume a [= b;
then b`"/\"a [= b`"/\"b by Th27;
then A1:    b`"/\"a [= Bottom L by Th47;
Bottom L [= b`"/\"a by Th41;
then b `"/\"a = Bottom L by A1,Th26;
hence b` [= a` by Th52;
end;