:: On the Closure Operator and the Closure System of Many Sorted Sets
:: by Artur Korni{\l}owicz
::
:: Received February 7, 1996
:: Copyright (c) 1996-2011 Association of Mizar Users


begin

notation
let I be set ;
let A, B be ManySortedSet of I;
synonym A in' B for A in B;
end;

notation
let I be set ;
let A, B be ManySortedSet of I;
synonym A c=' B for A c= B;
end;

theorem :: CLOSURE2:1
for M being non empty set
for X, Y being Element of M st X c= Y holds
(id M) . X c= (id M) . Y
proof end;

theorem :: CLOSURE2:2
canceled;

theorem Th3: :: CLOSURE2:3
for I being non empty set
for A being ManySortedSet of I
for B being ManySortedSubset of A holds rng B c= union (rng (bool A))
proof end;

begin

definition
let I be set ;
let M be ManySortedSet of I;
defpred S1[ set ] means $1 is ManySortedSubset of M;
func Bool M -> set means :Def1: :: CLOSURE2:def 1
for x being set holds
( x in it iff x is ManySortedSubset of M );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is ManySortedSubset of M )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is ManySortedSubset of M ) ) & ( for x being set holds
( x in b2 iff x is ManySortedSubset of M ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Bool CLOSURE2:def 1 :
for I being set
for M being ManySortedSet of I
for b3 being set holds
( b3 = Bool M iff for x being set holds
( x in b3 iff x is ManySortedSubset of M ) );

registration
let I be set ;
let M be ManySortedSet of I;
cluster Bool M -> non empty functional with_common_domain ;
coherence
( not Bool M is empty & Bool M is functional & Bool M is with_common_domain )
proof end;
end;

definition
let I be set ;
let M be ManySortedSet of I;
mode SubsetFamily of M is Subset of (Bool M);
end;

definition
let I be set ;
let M be ManySortedSet of I;
:: original: Bool
redefine func Bool M -> SubsetFamily of M;
coherence
Bool M is SubsetFamily of M
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster non empty functional with_common_domain Element of bool (Bool M);
existence
ex b1 being SubsetFamily of M st
( not b1 is empty & b1 is functional & b1 is with_common_domain )
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster empty functional finite Element of bool (Bool M);
existence
ex b1 being SubsetFamily of M st
( b1 is empty & b1 is finite )
proof end;
end;

definition
let I be set ;
let M be ManySortedSet of I;
let S be non empty SubsetFamily of M;
:: original: Element
redefine mode Element of S -> ManySortedSubset of M;
coherence
for b1 being Element of S holds b1 is ManySortedSubset of M
proof end;
end;

theorem Th4: :: CLOSURE2:4
for I being set
for M being ManySortedSet of I
for SF, SG being SubsetFamily of M holds SF \/ SG is SubsetFamily of M ;

theorem :: CLOSURE2:5
for I being set
for M being ManySortedSet of I
for SF, SG being SubsetFamily of M holds SF /\ SG is SubsetFamily of M ;

theorem :: CLOSURE2:6
for I, x being set
for M being ManySortedSet of I
for SF being SubsetFamily of M holds SF \ x is SubsetFamily of M ;

theorem :: CLOSURE2:7
for I being set
for M being ManySortedSet of I
for SF, SG being SubsetFamily of M holds SF \+\ SG is SubsetFamily of M ;

theorem Th8: :: CLOSURE2:8
for I being set
for A, M being ManySortedSet of I st A c= M holds
{A} is SubsetFamily of M
proof end;

theorem Th9: :: CLOSURE2:9
for I being set
for A, M, B being ManySortedSet of I st A c= M & B c= M holds
{A,B} is SubsetFamily of M
proof end;

theorem Th10: :: CLOSURE2:10
for I being set
for M being ManySortedSet of I
for E, T being Element of Bool M holds E /\ T in Bool M
proof end;

theorem Th11: :: CLOSURE2:11
for I being set
for M being ManySortedSet of I
for E, T being Element of Bool M holds E \/ T in Bool M
proof end;

theorem :: CLOSURE2:12
for I being set
for A, M being ManySortedSet of I
for E being Element of Bool M holds E \ A in Bool M
proof end;

theorem :: CLOSURE2:13
for I being set
for M being ManySortedSet of I
for E, T being Element of Bool M holds E \+\ T in Bool M
proof end;

begin

definition
let S be functional set ;
canceled;
func |.S.| -> Function means :Def3: :: CLOSURE2:def 3
ex A being non empty functional set st
( A = S & dom it = meet { (dom x) where x is Element of A : verum } & ( for i being set st i in dom it holds
it . i = { (x . i) where x is Element of A : verum } ) ) if S <> {}
otherwise it = {} ;
existence
( ( S <> {} implies ex b1 being Function ex A being non empty functional set st
( A = S & dom b1 = meet { (dom x) where x is Element of A : verum } & ( for i being set st i in dom b1 holds
b1 . i = { (x . i) where x is Element of A : verum } ) ) ) & ( not S <> {} implies ex b1 being Function st b1 = {} ) )
proof end;
uniqueness
for b1, b2 being Function holds
( ( S <> {} & ex A being non empty functional set st
( A = S & dom b1 = meet { (dom x) where x is Element of A : verum } & ( for i being set st i in dom b1 holds
b1 . i = { (x . i) where x is Element of A : verum } ) ) & ex A being non empty functional set st
( A = S & dom b2 = meet { (dom x) where x is Element of A : verum } & ( for i being set st i in dom b2 holds
b2 . i = { (x . i) where x is Element of A : verum } ) ) implies b1 = b2 ) & ( not S <> {} & b1 = {} & b2 = {} implies b1 = b2 ) )
proof end;
consistency
for b1 being Function holds verum
;
end;

:: deftheorem CLOSURE2:def 2 :
canceled;

:: deftheorem Def3 defines |. CLOSURE2:def 3 :
for S being functional set
for b2 being Function holds
( ( S <> {} implies ( b2 = |.S.| iff ex A being non empty functional set st
( A = S & dom b2 = meet { (dom x) where x is Element of A : verum } & ( for i being set st i in dom b2 holds
b2 . i = { (x . i) where x is Element of A : verum } ) ) ) ) & ( not S <> {} implies ( b2 = |.S.| iff b2 = {} ) ) );

theorem Th14: :: CLOSURE2:14
for I being set
for M being ManySortedSet of I
for SF being non empty SubsetFamily of M holds dom |.SF.| = I
proof end;

registration
let S be empty functional set ;
cluster |.S.| -> empty ;
coherence
|.S.| is empty
by Def3;
end;

definition
let I be set ;
let M be ManySortedSet of I;
let S be SubsetFamily of M;
func |:S:| -> ManySortedSet of I equals :Def4: :: CLOSURE2:def 4
|.S.| if S <> {}
otherwise [[0]] I;
coherence
( ( S <> {} implies |.S.| is ManySortedSet of I ) & ( not S <> {} implies [[0]] I is ManySortedSet of I ) )
proof end;
consistency
for b1 being ManySortedSet of I holds verum
;
end;

:: deftheorem Def4 defines |: CLOSURE2:def 4 :
for I being set
for M being ManySortedSet of I
for S being SubsetFamily of M holds
( ( S <> {} implies |:S:| = |.S.| ) & ( not S <> {} implies |:S:| = [[0]] I ) );

registration
let I be set ;
let M be ManySortedSet of I;
let S be empty SubsetFamily of M;
cluster |:S:| -> V9() ;
coherence
|:S:| is empty-yielding
proof end;
end;

theorem Th15: :: CLOSURE2:15
for I being set
for M being ManySortedSet of I
for SF being SubsetFamily of M st not SF is empty holds
for i being set st i in I holds
|:SF:| . i = { (x . i) where x is Element of Bool M : x in SF }
proof end;

registration
let I be set ;
let M be ManySortedSet of I;
let SF be non empty SubsetFamily of M;
cluster |:SF:| -> V8() ;
coherence
|:SF:| is non-empty
proof end;
end;

theorem Th16: :: CLOSURE2:16
for f being Function holds dom |.{f}.| = dom f
proof end;

theorem :: CLOSURE2:17
for f, f1 being Function holds dom |.{f,f1}.| = (dom f) /\ (dom f1)
proof end;

theorem Th18: :: CLOSURE2:18
for i being set
for f being Function st i in dom f holds
|.{f}.| . i = {(f . i)}
proof end;

theorem :: CLOSURE2:19
for i, I being set
for M being ManySortedSet of I
for f being Function
for SF being SubsetFamily of M st i in I & SF = {f} holds
|:SF:| . i = {(f . i)}
proof end;

theorem Th20: :: CLOSURE2:20
for i being set
for f, f1 being Function st i in dom |.{f,f1}.| holds
|.{f,f1}.| . i = {(f . i),(f1 . i)}
proof end;

theorem Th21: :: CLOSURE2:21
for i, I being set
for M being ManySortedSet of I
for f, f1 being Function
for SF being SubsetFamily of M st i in I & SF = {f,f1} holds
|:SF:| . i = {(f . i),(f1 . i)}
proof end;

definition
let I be set ;
let M be ManySortedSet of I;
let SF be SubsetFamily of M;
:: original: |:
redefine func |:SF:| -> MSSubsetFamily of M;
coherence
|:SF:| is MSSubsetFamily of M
proof end;
end;

theorem Th22: :: CLOSURE2:22
for I being set
for M, A being ManySortedSet of I
for SF being SubsetFamily of M st A in SF holds
A in' |:SF:|
proof end;

theorem Th23: :: CLOSURE2:23
for I being set
for M, A, B being ManySortedSet of I
for SF being SubsetFamily of M st SF = {A,B} holds
union |:SF:| = A \/ B
proof end;

theorem Th24: :: CLOSURE2:24
for I being set
for M being ManySortedSet of I
for SF being SubsetFamily of M
for E, T being Element of Bool M st SF = {E,T} holds
meet |:SF:| = E /\ T
proof end;

theorem Th25: :: CLOSURE2:25
for I being set
for M being ManySortedSet of I
for SF being SubsetFamily of M
for Z being ManySortedSubset of M st ( for Z1 being ManySortedSet of I st Z1 in SF holds
Z c=' Z1 ) holds
Z c=' meet |:SF:|
proof end;

theorem :: CLOSURE2:26
for I being set
for M being ManySortedSet of I holds |:(Bool M):| = bool M
proof end;

definition
let I be set ;
let M be ManySortedSet of I;
let IT be SubsetFamily of M;
attr IT is additive means :: CLOSURE2:def 5
for A, B being ManySortedSet of I st A in IT & B in IT holds
A \/ B in IT;
attr IT is absolutely-additive means :Def6: :: CLOSURE2:def 6
for F being SubsetFamily of M st F c= IT holds
union |:F:| in IT;
attr IT is multiplicative means :: CLOSURE2:def 7
for A, B being ManySortedSet of I st A in IT & B in IT holds
A /\ B in IT;
attr IT is absolutely-multiplicative means :Def8: :: CLOSURE2:def 8
for F being SubsetFamily of M st F c= IT holds
meet |:F:| in IT;
attr IT is properly-upper-bound means :Def9: :: CLOSURE2:def 9
M in IT;
attr IT is properly-lower-bound means :Def10: :: CLOSURE2:def 10
[[0]] I in IT;
end;

:: deftheorem defines additive CLOSURE2:def 5 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is additive iff for A, B being ManySortedSet of I st A in IT & B in IT holds
A \/ B in IT );

:: deftheorem Def6 defines absolutely-additive CLOSURE2:def 6 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is absolutely-additive iff for F being SubsetFamily of M st F c= IT holds
union |:F:| in IT );

:: deftheorem defines multiplicative CLOSURE2:def 7 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is multiplicative iff for A, B being ManySortedSet of I st A in IT & B in IT holds
A /\ B in IT );

:: deftheorem Def8 defines absolutely-multiplicative CLOSURE2:def 8 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is absolutely-multiplicative iff for F being SubsetFamily of M st F c= IT holds
meet |:F:| in IT );

:: deftheorem Def9 defines properly-upper-bound CLOSURE2:def 9 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is properly-upper-bound iff M in IT );

:: deftheorem Def10 defines properly-lower-bound CLOSURE2:def 10 :
for I being set
for M being ManySortedSet of I
for IT being SubsetFamily of M holds
( IT is properly-lower-bound iff [[0]] I in IT );

Lm1: for I being set
for M being ManySortedSet of I holds
( Bool M is additive & Bool M is absolutely-additive & Bool M is multiplicative & Bool M is absolutely-multiplicative & Bool M is properly-upper-bound & Bool M is properly-lower-bound )
proof end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster non empty functional with_common_domain additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound Element of bool (Bool M);
existence
ex b1 being SubsetFamily of M st
( not b1 is empty & b1 is functional & b1 is with_common_domain & b1 is additive & b1 is absolutely-additive & b1 is multiplicative & b1 is absolutely-multiplicative & b1 is properly-upper-bound & b1 is properly-lower-bound )
proof end;
end;

definition
let I be set ;
let M be ManySortedSet of I;
:: original: Bool
redefine func Bool M -> additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound SubsetFamily of M;
coherence
Bool M is additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound SubsetFamily of M
by Lm1;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster absolutely-additive -> additive Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is absolutely-additive holds
b1 is additive
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster absolutely-multiplicative -> multiplicative Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is absolutely-multiplicative holds
b1 is multiplicative
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster absolutely-multiplicative -> properly-upper-bound Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is absolutely-multiplicative holds
b1 is properly-upper-bound
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster properly-upper-bound -> non empty Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is properly-upper-bound holds
not b1 is empty
by Def9;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster absolutely-additive -> properly-lower-bound Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is absolutely-additive holds
b1 is properly-lower-bound
proof end;
end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster properly-lower-bound -> non empty Element of bool (Bool M);
coherence
for b1 being SubsetFamily of M st b1 is properly-lower-bound holds
not b1 is empty
by Def10;
end;

begin

definition
let I be set ;
let M be ManySortedSet of I;
mode SetOp of M is Function of (Bool M),(Bool M);
end;

definition
let I be set ;
let M be ManySortedSet of I;
let f be SetOp of M;
let x be Element of Bool M;
:: original: .
redefine func f . x -> Element of Bool M;
coherence
f . x is Element of Bool M
proof end;
end;

definition
let I be set ;
let M be ManySortedSet of I;
let IT be SetOp of M;
canceled;
attr IT is reflexive means :Def12: :: CLOSURE2:def 12
for x being Element of Bool M holds x c=' IT . x;
attr IT is monotonic means :Def13: :: CLOSURE2:def 13
for x, y being Element of Bool M st x c=' y holds
IT . x c=' IT . y;
attr IT is idempotent means :Def14: :: CLOSURE2:def 14
for x being Element of Bool M holds IT . x = IT . (IT . x);
attr IT is topological means :Def15: :: CLOSURE2:def 15
for x, y being Element of Bool M holds IT . (x \/ y) = (IT . x) \/ (IT . y);
end;

:: deftheorem CLOSURE2:def 11 :
canceled;

:: deftheorem Def12 defines reflexive CLOSURE2:def 12 :
for I being set
for M being ManySortedSet of I
for IT being SetOp of M holds
( IT is reflexive iff for x being Element of Bool M holds x c=' IT . x );

:: deftheorem Def13 defines monotonic CLOSURE2:def 13 :
for I being set
for M being ManySortedSet of I
for IT being SetOp of M holds
( IT is monotonic iff for x, y being Element of Bool M st x c=' y holds
IT . x c=' IT . y );

:: deftheorem Def14 defines idempotent CLOSURE2:def 14 :
for I being set
for M being ManySortedSet of I
for IT being SetOp of M holds
( IT is idempotent iff for x being Element of Bool M holds IT . x = IT . (IT . x) );

:: deftheorem Def15 defines topological CLOSURE2:def 15 :
for I being set
for M being ManySortedSet of I
for IT being SetOp of M holds
( IT is topological iff for x, y being Element of Bool M holds IT . (x \/ y) = (IT . x) \/ (IT . y) );

registration
let I be set ;
let M be ManySortedSet of I;
cluster non empty Relation-like Function-like V17( Bool M) quasi_total reflexive monotonic idempotent topological Element of bool [:(Bool M),(Bool M):];
existence
ex b1 being SetOp of M st
( b1 is reflexive & b1 is monotonic & b1 is idempotent & b1 is topological )
proof end;
end;

theorem :: CLOSURE2:27
for I being set
for A being ManySortedSet of I holds id (Bool A) is reflexive SetOp of A
proof end;

theorem :: CLOSURE2:28
for I being set
for A being ManySortedSet of I holds id (Bool A) is monotonic SetOp of A
proof end;

theorem :: CLOSURE2:29
for I being set
for A being ManySortedSet of I holds id (Bool A) is idempotent SetOp of A
proof end;

theorem :: CLOSURE2:30
for I being set
for A being ManySortedSet of I holds id (Bool A) is topological SetOp of A
proof end;

theorem :: CLOSURE2:31
for I being set
for M being ManySortedSet of I
for E being Element of Bool M
for g being SetOp of M st E = M & g is reflexive holds
E = g . E
proof end;

theorem :: CLOSURE2:32
for I being set
for M being ManySortedSet of I
for g being SetOp of M st g is reflexive & ( for X being Element of Bool M holds g . X c= X ) holds
g is idempotent
proof end;

theorem :: CLOSURE2:33
for I being set
for M being ManySortedSet of I
for E, T being Element of Bool M
for g being SetOp of M
for A being Element of Bool M st A = E /\ T & g is monotonic holds
g . A c= (g . E) /\ (g . T)
proof end;

registration
let I be set ;
let M be ManySortedSet of I;
cluster Function-like quasi_total topological -> monotonic Element of bool [:(Bool M),(Bool M):];
coherence
for b1 being SetOp of M st b1 is topological holds
b1 is monotonic
proof end;
end;

theorem :: CLOSURE2:34
for I being set
for M being ManySortedSet of I
for E, T being Element of Bool M
for g being SetOp of M
for A being Element of Bool M st A = E \ T & g is topological holds
(g . E) \ (g . T) c= g . A
proof end;

definition
let I be set ;
let M be ManySortedSet of I;
let h, g be SetOp of M;
:: original: *
redefine func g * h -> SetOp of M;
coherence
h * g is SetOp of M
proof end;
end;

theorem :: CLOSURE2:35
for I being set
for M being ManySortedSet of I
for g, h being SetOp of M st g is reflexive & h is reflexive holds
g * h is reflexive
proof end;

theorem :: CLOSURE2:36
for I being set
for M being ManySortedSet of I
for g, h being SetOp of M st g is monotonic & h is monotonic holds
g * h is monotonic
proof end;

theorem :: CLOSURE2:37
for I being set
for M being ManySortedSet of I
for g, h being SetOp of M st g is idempotent & h is idempotent & g * h = h * g holds
g * h is idempotent
proof end;

theorem :: CLOSURE2:38
for I being set
for M being ManySortedSet of I
for g, h being SetOp of M st g is topological & h is topological holds
g * h is topological
proof end;

begin

definition
let S be 1-sorted ;
attr c2 is strict ;
struct ClosureStr of S -> many-sorted of S;
aggr ClosureStr(# Sorts, Family #) -> ClosureStr of S;
sel Family c2 -> SubsetFamily of the Sorts of c2;
end;

definition
let S be 1-sorted ;
let IT be ClosureStr of S;
attr IT is additive means :Def16: :: CLOSURE2:def 16
the Family of IT is additive ;
attr IT is absolutely-additive means :Def17: :: CLOSURE2:def 17
the Family of IT is absolutely-additive ;
attr IT is multiplicative means :Def18: :: CLOSURE2:def 18
the Family of IT is multiplicative ;
attr IT is absolutely-multiplicative means :Def19: :: CLOSURE2:def 19
the Family of IT is absolutely-multiplicative ;
attr IT is properly-upper-bound means :Def20: :: CLOSURE2:def 20
the Family of IT is properly-upper-bound ;
attr IT is properly-lower-bound means :Def21: :: CLOSURE2:def 21
the Family of IT is properly-lower-bound ;
end;

:: deftheorem Def16 defines additive CLOSURE2:def 16 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is additive iff the Family of IT is additive );

:: deftheorem Def17 defines absolutely-additive CLOSURE2:def 17 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is absolutely-additive iff the Family of IT is absolutely-additive );

:: deftheorem Def18 defines multiplicative CLOSURE2:def 18 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is multiplicative iff the Family of IT is multiplicative );

:: deftheorem Def19 defines absolutely-multiplicative CLOSURE2:def 19 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is absolutely-multiplicative iff the Family of IT is absolutely-multiplicative );

:: deftheorem Def20 defines properly-upper-bound CLOSURE2:def 20 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is properly-upper-bound iff the Family of IT is properly-upper-bound );

:: deftheorem Def21 defines properly-lower-bound CLOSURE2:def 21 :
for S being 1-sorted
for IT being ClosureStr of S holds
( IT is properly-lower-bound iff the Family of IT is properly-lower-bound );

definition
let S be 1-sorted ;
let MS be many-sorted of S;
func Full MS -> ClosureStr of S equals :: CLOSURE2:def 22
ClosureStr(# the Sorts of MS,(Bool the Sorts of MS) #);
correctness
coherence
ClosureStr(# the Sorts of MS,(Bool the Sorts of MS) #) is ClosureStr of S
;
;
end;

:: deftheorem defines Full CLOSURE2:def 22 :
for S being 1-sorted
for MS being many-sorted of S holds Full MS = ClosureStr(# the Sorts of MS,(Bool the Sorts of MS) #);

registration
let S be 1-sorted ;
let MS be many-sorted of S;
cluster Full MS -> strict additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound ;
coherence
( Full MS is strict & Full MS is additive & Full MS is absolutely-additive & Full MS is multiplicative & Full MS is absolutely-multiplicative & Full MS is properly-upper-bound & Full MS is properly-lower-bound )
proof end;
end;

registration
let S be 1-sorted ;
let MS be non-empty many-sorted of S;
cluster Full MS -> non-empty ;
coherence
Full MS is non-empty
by MSUALG_1:def 8;
end;

registration
let S be 1-sorted ;
cluster non-empty strict additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound ClosureStr of S;
existence
ex b1 being ClosureStr of S st
( b1 is strict & b1 is non-empty & b1 is additive & b1 is absolutely-additive & b1 is multiplicative & b1 is absolutely-multiplicative & b1 is properly-upper-bound & b1 is properly-lower-bound )
proof end;
end;

registration
let S be 1-sorted ;
let CS be additive ClosureStr of S;
cluster the Family of CS -> additive ;
coherence
the Family of CS is additive
by Def16;
end;

registration
let S be 1-sorted ;
let CS be absolutely-additive ClosureStr of S;
cluster the Family of CS -> absolutely-additive ;
coherence
the Family of CS is absolutely-additive
by Def17;
end;

registration
let S be 1-sorted ;
let CS be multiplicative ClosureStr of S;
cluster the Family of CS -> multiplicative ;
coherence
the Family of CS is multiplicative
by Def18;
end;

registration
let S be 1-sorted ;
let CS be absolutely-multiplicative ClosureStr of S;
cluster the Family of CS -> absolutely-multiplicative ;
coherence
the Family of CS is absolutely-multiplicative
by Def19;
end;

registration
let S be 1-sorted ;
let CS be properly-upper-bound ClosureStr of S;
cluster the Family of CS -> properly-upper-bound ;
coherence
the Family of CS is properly-upper-bound
by Def20;
end;

registration
let S be 1-sorted ;
let CS be properly-lower-bound ClosureStr of S;
cluster the Family of CS -> properly-lower-bound ;
coherence
the Family of CS is properly-lower-bound
by Def21;
end;

registration
let S be 1-sorted ;
let M be V8() ManySortedSet of the carrier of S;
let F be SubsetFamily of M;
cluster ClosureStr(# M,F #) -> non-empty ;
coherence
ClosureStr(# M,F #) is non-empty
proof end;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be additive SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> additive ;
coherence
ClosureStr(# the Sorts of MS,F #) is additive
by Def16;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be absolutely-additive SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> absolutely-additive ;
coherence
ClosureStr(# the Sorts of MS,F #) is absolutely-additive
by Def17;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be multiplicative SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> multiplicative ;
coherence
ClosureStr(# the Sorts of MS,F #) is multiplicative
by Def18;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be absolutely-multiplicative SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> absolutely-multiplicative ;
coherence
ClosureStr(# the Sorts of MS,F #) is absolutely-multiplicative
by Def19;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be properly-upper-bound SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> properly-upper-bound ;
coherence
ClosureStr(# the Sorts of MS,F #) is properly-upper-bound
by Def20;
end;

registration
let S be 1-sorted ;
let MS be many-sorted of S;
let F be properly-lower-bound SubsetFamily of the Sorts of MS;
cluster ClosureStr(# the Sorts of MS,F #) -> properly-lower-bound ;
coherence
ClosureStr(# the Sorts of MS,F #) is properly-lower-bound
by Def21;
end;

registration
let S be 1-sorted ;
cluster absolutely-additive -> additive ClosureStr of S;
coherence
for b1 being ClosureStr of S st b1 is absolutely-additive holds
b1 is additive
proof end;
end;

registration
let S be 1-sorted ;
cluster absolutely-multiplicative -> multiplicative ClosureStr of S;
coherence
for b1 being ClosureStr of S st b1 is absolutely-multiplicative holds
b1 is multiplicative
proof end;
end;

registration
let S be 1-sorted ;
cluster absolutely-multiplicative -> properly-upper-bound ClosureStr of S;
coherence
for b1 being ClosureStr of S st b1 is absolutely-multiplicative holds
b1 is properly-upper-bound
proof end;
end;

registration
let S be 1-sorted ;
cluster absolutely-additive -> properly-lower-bound ClosureStr of S;
coherence
for b1 being ClosureStr of S st b1 is absolutely-additive holds
b1 is properly-lower-bound
proof end;
end;

definition
let S be 1-sorted ;
mode ClosureSystem of S is absolutely-multiplicative ClosureStr of S;
end;

definition
let I be set ;
let M be ManySortedSet of I;
mode ClosureOperator of M is reflexive monotonic idempotent SetOp of M;
end;

definition
let S be 1-sorted ;
let A be ManySortedSet of the carrier of S;
let g be ClosureOperator of A;
func ClOp->ClSys g -> strict ClosureStr of S means :Def23: :: CLOSURE2:def 23
( the Sorts of it = A & the Family of it = { x where x is Element of Bool A : g . x = x } );
existence
ex b1 being strict ClosureStr of S st
( the Sorts of b1 = A & the Family of b1 = { x where x is Element of Bool A : g . x = x } )
proof end;
uniqueness
for b1, b2 being strict ClosureStr of S st the Sorts of b1 = A & the Family of b1 = { x where x is Element of Bool A : g . x = x } & the Sorts of b2 = A & the Family of b2 = { x where x is Element of Bool A : g . x = x } holds
b1 = b2
;
end;

:: deftheorem Def23 defines ClOp->ClSys CLOSURE2:def 23 :
for S being 1-sorted
for A being ManySortedSet of the carrier of S
for g being ClosureOperator of A
for b4 being strict ClosureStr of S holds
( b4 = ClOp->ClSys g iff ( the Sorts of b4 = A & the Family of b4 = { x where x is Element of Bool A : g . x = x } ) );

registration
let S be 1-sorted ;
let A be ManySortedSet of the carrier of S;
let g be ClosureOperator of A;
cluster ClOp->ClSys g -> strict absolutely-multiplicative ;
coherence
ClOp->ClSys g is absolutely-multiplicative
proof end;
end;

definition
let S be 1-sorted ;
let A be ClosureSystem of S;
let C be ManySortedSubset of the Sorts of A;
func Cl C -> Element of Bool the Sorts of A means :Def24: :: CLOSURE2:def 24
ex F being SubsetFamily of the Sorts of A st
( it = meet |:F:| & F = { X where X is Element of Bool the Sorts of A : ( C c=' X & X in the Family of A ) } );
existence
ex b1 being Element of Bool the Sorts of A ex F being SubsetFamily of the Sorts of A st
( b1 = meet |:F:| & F = { X where X is Element of Bool the Sorts of A : ( C c=' X & X in the Family of A ) } )
proof end;
uniqueness
for b1, b2 being Element of Bool the Sorts of A st ex F being SubsetFamily of the Sorts of A st
( b1 = meet |:F:| & F = { X where X is Element of Bool the Sorts of A : ( C c=' X & X in the Family of A ) } ) & ex F being SubsetFamily of the Sorts of A st
( b2 = meet |:F:| & F = { X where X is Element of Bool the Sorts of A : ( C c=' X & X in the Family of A ) } ) holds
b1 = b2
;
end;

:: deftheorem Def24 defines Cl CLOSURE2:def 24 :
for S being 1-sorted
for A being ClosureSystem of S
for C being ManySortedSubset of the Sorts of A
for b4 being Element of Bool the Sorts of A holds
( b4 = Cl C iff ex F being SubsetFamily of the Sorts of A st
( b4 = meet |:F:| & F = { X where X is Element of Bool the Sorts of A : ( C c=' X & X in the Family of A ) } ) );

theorem :: CLOSURE2:39
canceled;

theorem Th40: :: CLOSURE2:40
for S being 1-sorted
for D being ClosureSystem of S
for a being Element of Bool the Sorts of D
for f being SetOp of the Sorts of D st a in the Family of D & ( for x being Element of Bool the Sorts of D holds f . x = Cl x ) holds
f . a = a
proof end;

theorem :: CLOSURE2:41
for S being 1-sorted
for D being ClosureSystem of S
for a being Element of Bool the Sorts of D
for f being SetOp of the Sorts of D st f . a = a & ( for x being Element of Bool the Sorts of D holds f . x = Cl x ) holds
a in the Family of D
proof end;

theorem Th42: :: CLOSURE2:42
for S being 1-sorted
for D being ClosureSystem of S
for f being SetOp of the Sorts of D st ( for x being Element of Bool the Sorts of D holds f . x = Cl x ) holds
( f is reflexive & f is monotonic & f is idempotent )
proof end;

definition
let S be 1-sorted ;
let D be ClosureSystem of S;
func ClSys->ClOp D -> ClosureOperator of the Sorts of D means :Def25: :: CLOSURE2:def 25
for x being Element of Bool the Sorts of D holds it . x = Cl x;
existence
ex b1 being ClosureOperator of the Sorts of D st
for x being Element of Bool the Sorts of D holds b1 . x = Cl x
proof end;
uniqueness
for b1, b2 being ClosureOperator of the Sorts of D st ( for x being Element of Bool the Sorts of D holds b1 . x = Cl x ) & ( for x being Element of Bool the Sorts of D holds b2 . x = Cl x ) holds
b1 = b2
proof end;
end;

:: deftheorem Def25 defines ClSys->ClOp CLOSURE2:def 25 :
for S being 1-sorted
for D being ClosureSystem of S
for b3 being ClosureOperator of the Sorts of D holds
( b3 = ClSys->ClOp D iff for x being Element of Bool the Sorts of D holds b3 . x = Cl x );

theorem :: CLOSURE2:43
for S being 1-sorted
for A being ManySortedSet of the carrier of S
for f being ClosureOperator of A holds ClSys->ClOp (ClOp->ClSys f) = f
proof end;

deffunc H1( set ) -> set = $1;

theorem :: CLOSURE2:44
for S being 1-sorted
for D being ClosureSystem of S holds ClOp->ClSys (ClSys->ClOp D) = ClosureStr(# the Sorts of D, the Family of D #)
proof end;