:: Algebra of Normal Forms Is a Heyting Algebra
:: by Andrzej Trybulec
::
:: Received January 3, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

theorem Th1: :: HEYTING1:1

for A, B, C being non empty set
for f being Function of A,B st ( for x being Element of A holds f . x in C ) holds
f is Function of A,C

proof end;

definition

let A be non empty set ;
let B, C be Element of Fin A;
:: original: c=
redefine pred B c= C means :: HEYTING1:def 1
for a being Element of A st a in B holds
a in C;
compatibility
( B c= C iff for a being Element of A st a in B holds
a in C )

proof end;

end;

:: deftheorem defines c= HEYTING1:def 1 :
for A being non empty set
for B, C being Element of Fin A holds
( B c= C iff for a being Element of A st a in B holds
a in C );

definition

let A be set ;
assume A1: not A is empty ;
func [A] -> non empty set equals :Def2: :: HEYTING1:def 2
A;
correctness
coherence
A is non empty set ;
by A1;

end;

:: deftheorem Def2 defines [ HEYTING1:def 2 :
for A being set st not A is empty holds
[A] = A;

theorem :: HEYTING1:2

canceled;

theorem :: HEYTING1:3

for A being set
for B being Element of Fin (DISJOINT_PAIRS A) st B = {} holds
mi B = {} by NORMFORM:64, XBOOLE_1:3;

Lm1: now

let A be set ; :: thesis: for a being Element of DISJOINT_PAIRS A holds {a} is Element of Normal_forms_on A
let a be Element of DISJOINT_PAIRS A; :: thesis: {a} is Element of Normal_forms_on A
reconsider B = {.a.} as Element of Fin (DISJOINT_PAIRS A) ;

now

let c, b be Element of DISJOINT_PAIRS A; :: thesis: ( c in B & b in B & c c= b implies c = b )
assume that
A1: c in B and
A2: b in B and
c c= b ; :: thesis: c = b
c = a by A1, TARSKI:def 1;
hence c = b by A2, TARSKI:def 1; :: thesis: verum

end;

hence {a} is Element of Normal_forms_on A by NORMFORM:53; :: thesis: verum

end;

registration

let A be set ;
cluster non empty Element of Normal_forms_on A;
existence
not for b₁ being Element of Normal_forms_on A holds b₁ is empty

proof end;

end;

definition

let A be set ;
let a be Element of DISJOINT_PAIRS A;
:: original: {
redefine func {a} -> Element of Normal_forms_on A;
coherence
{a} is Element of Normal_forms_on A by Lm1;

end;

definition

let A be set ;
let u be Element of (NormForm A);
func @ u -> Element of Normal_forms_on A equals :: HEYTING1:def 3
u;
coherence
u is Element of Normal_forms_on A by NORMFORM:def 14;

end;

:: deftheorem defines @ HEYTING1:def 3 :
for A being set
for u being Element of (NormForm A) holds @ u = u;

theorem :: HEYTING1:4

canceled;

theorem :: HEYTING1:5

canceled;

theorem :: HEYTING1:6

canceled;

theorem Th7: :: HEYTING1:7

for A being set
for K being Element of Normal_forms_on A holds mi (K ^ K) = K

proof end;

theorem Th8: :: HEYTING1:8

for A being set
for K being Element of Normal_forms_on A
for X being set st X c= K holds
X in Normal_forms_on A

proof end;

theorem :: HEYTING1:9

canceled;

theorem Th10: :: HEYTING1:10

for A being set
for u being Element of (NormForm A)
for X being set st X c= u holds
X is Element of (NormForm A)

proof end;

definition

let A be set ;
func Atom A -> Function of (DISJOINT_PAIRS A), the carrier of (NormForm A) means :Def4: :: HEYTING1:def 4
for a being Element of DISJOINT_PAIRS A holds it . a = {a};
existence
ex b₁ being Function of (DISJOINT_PAIRS A), the carrier of (NormForm A) st
for a being Element of DISJOINT_PAIRS A holds b₁ . a = {a}

proof end;

uniqueness
for b₁, b₂ being Function of (DISJOINT_PAIRS A), the carrier of (NormForm A) st ( for a being Element of DISJOINT_PAIRS A holds b₁ . a = {a} ) & ( for a being Element of DISJOINT_PAIRS A holds b₂ . a = {a} ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Atom HEYTING1:def 4 :
for A being set
for b₂ being Function of (DISJOINT_PAIRS A), the carrier of (NormForm A) holds
( b₂ = Atom A iff for a being Element of DISJOINT_PAIRS A holds b₂ . a = {a} );

theorem Th11: :: HEYTING1:11

for A being set
for c, a being Element of DISJOINT_PAIRS A st c in (Atom A) . a holds
c = a

proof end;

theorem Th12: :: HEYTING1:12

for A being set
for a being Element of DISJOINT_PAIRS A holds a in (Atom A) . a

proof end;

theorem :: HEYTING1:13

for A being set
for a being Element of DISJOINT_PAIRS A holds (Atom A) . a = (singleton (DISJOINT_PAIRS A)) . a

proof end;

theorem Th14: :: HEYTING1:14

for A being set
for K being Element of Normal_forms_on A holds FinJoin (K,(Atom A)) = FinUnion (K,(singleton (DISJOINT_PAIRS A)))

proof end;

theorem Th15: :: HEYTING1:15

for A being set
for u being Element of (NormForm A) holds u = FinJoin ((@ u),(Atom A))

proof end;

Lm2: for A being set
for u, v being Element of (NormForm A) st u [= v holds
for x being Element of [:(Fin A),(Fin A):] st x in u holds
ex b being Element of DISJOINT_PAIRS A st
( b in v & b c= x )

proof end;

Lm3: for A being set
for u, v being Element of (NormForm A) st ( for a being Element of DISJOINT_PAIRS A st a in u holds
ex b being Element of DISJOINT_PAIRS A st
( b in v & b c= a ) ) holds
u [= v

proof end;

definition

let A be set ;
func pair_diff A -> BinOp of [:(Fin A),(Fin A):] means :Def5: :: HEYTING1:def 5
for a, b being Element of [:(Fin A),(Fin A):] holds it . (a,b) = a \ b;
existence
ex b₁ being BinOp of [:(Fin A),(Fin A):] st
for a, b being Element of [:(Fin A),(Fin A):] holds b₁ . (a,b) = a \ b

proof end;

correctness
uniqueness
for b₁, b₂ being BinOp of [:(Fin A),(Fin A):] st ( for a, b being Element of [:(Fin A),(Fin A):] holds b₁ . (a,b) = a \ b ) & ( for a, b being Element of [:(Fin A),(Fin A):] holds b₂ . (a,b) = a \ b ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def5 defines pair_diff HEYTING1:def 5 :
for A being set
for b₂ being BinOp of [:(Fin A),(Fin A):] holds
( b₂ = pair_diff A iff for a, b being Element of [:(Fin A),(Fin A):] holds b₂ . (a,b) = a \ b );

definition

let A be set ;
let B be Element of Fin (DISJOINT_PAIRS A);
func - B -> Element of Fin (DISJOINT_PAIRS A) equals :: HEYTING1:def 6
(DISJOINT_PAIRS A) /\ { [ { (g . t1) where t1 is Element of DISJOINT_PAIRS A : ( g . t1 in t1 `2 & t1 in B ) } , { (g . t2) where t2 is Element of DISJOINT_PAIRS A : ( g . t2 in t2 `1 & t2 in B ) } ] where g is Element of Funcs ((DISJOINT_PAIRS A),[A]) : for s being Element of DISJOINT_PAIRS A st s in B holds
g . s in (s `1) \/ (s `2) } ;
coherence
(DISJOINT_PAIRS A) /\ { [ { (g . t1) where t1 is Element of DISJOINT_PAIRS A : ( g . t1 in t1 `2 & t1 in B ) } , { (g . t2) where t2 is Element of DISJOINT_PAIRS A : ( g . t2 in t2 `1 & t2 in B ) } ] where g is Element of Funcs ((DISJOINT_PAIRS A),[A]) : for s being Element of DISJOINT_PAIRS A st s in B holds
g . s in (s `1) \/ (s `2) } is Element of Fin (DISJOINT_PAIRS A)

proof end;

correctness
;
let C be Element of Fin (DISJOINT_PAIRS A);
func B =>> C -> Element of Fin (DISJOINT_PAIRS A) equals :: HEYTING1:def 7
(DISJOINT_PAIRS A) /\ { (FinPairUnion (B,((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))))) where f is Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) : f .: B c= C } ;
coherence
(DISJOINT_PAIRS A) /\ { (FinPairUnion (B,((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))))) where f is Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) : f .: B c= C } is Element of Fin (DISJOINT_PAIRS A)

proof end;

correctness
;

end;

:: deftheorem defines - HEYTING1:def 6 :
for A being set
for B being Element of Fin (DISJOINT_PAIRS A) holds - B = (DISJOINT_PAIRS A) /\ { [ { (g . t1) where t1 is Element of DISJOINT_PAIRS A : ( g . t1 in t1 `2 & t1 in B ) } , { (g . t2) where t2 is Element of DISJOINT_PAIRS A : ( g . t2 in t2 `1 & t2 in B ) } ] where g is Element of Funcs ((DISJOINT_PAIRS A),[A]) : for s being Element of DISJOINT_PAIRS A st s in B holds
g . s in (s `1) \/ (s `2) } ;

:: deftheorem defines =>> HEYTING1:def 7 :
for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds B =>> C = (DISJOINT_PAIRS A) /\ { (FinPairUnion (B,((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))))) where f is Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) : f .: B c= C } ;

theorem Th16: :: HEYTING1:16

for A being set
for B being Element of Fin (DISJOINT_PAIRS A)
for c being Element of DISJOINT_PAIRS A st c in - B holds
ex g being Element of Funcs ((DISJOINT_PAIRS A),[A]) st
( ( for s being Element of DISJOINT_PAIRS A st s in B holds
g . s in (s `1) \/ (s `2) ) & c = [ { (g . t1) where t1 is Element of DISJOINT_PAIRS A : ( g . t1 in t1 `2 & t1 in B ) } , { (g . t2) where t2 is Element of DISJOINT_PAIRS A : ( g . t2 in t2 `1 & t2 in B ) } ] )

proof end;

theorem Th17: :: HEYTING1:17

for A being set holds [{},{}] is Element of DISJOINT_PAIRS A

proof end;

theorem Th18: :: HEYTING1:18

for A being set
for K being Element of Normal_forms_on A st K = {} holds
- K = {[{},{}]}

proof end;

theorem Th19: :: HEYTING1:19

for A being set
for K, L being Element of Normal_forms_on A st K = {} & L = {} holds
K =>> L = {[{},{}]}

proof end;

theorem Th20: :: HEYTING1:20

for a being Element of DISJOINT_PAIRS {} holds a = [{},{}]

proof end;

theorem Th21: :: HEYTING1:21

DISJOINT_PAIRS {} = {[{},{}]}

proof end;

Lm4: Fin (DISJOINT_PAIRS {}) = {{},{[{},{}]}}

proof end;

theorem Th22: :: HEYTING1:22

for A being set holds {[{},{}]} is Element of Normal_forms_on A

proof end;

theorem Th23: :: HEYTING1:23

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A)
for c being Element of DISJOINT_PAIRS A st c in B =>> C holds
ex f being Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) st
( f .: B c= C & c = FinPairUnion (B,((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A))))) )

proof end;

theorem Th24: :: HEYTING1:24

for A being set
for a being Element of DISJOINT_PAIRS A
for K being Element of Normal_forms_on A st K ^ {a} = {} holds
ex b being Element of DISJOINT_PAIRS A st
( b in - K & b c= a )

proof end;

Lm5: now

let A be set ; :: thesis: for K being Element of Normal_forms_on A
for b being Element of DISJOINT_PAIRS A
for f being Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) holds ((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))) . b = (f . b) \ b
let K be Element of Normal_forms_on A; :: thesis: for b being Element of DISJOINT_PAIRS A
for f being Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) holds ((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))) . b = (f . b) \ b
let b be Element of DISJOINT_PAIRS A; :: thesis: for f being Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]) holds ((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))) . b = (f . b) \ b
let f be Element of Funcs ((DISJOINT_PAIRS A),[:(Fin A),(Fin A):]); :: thesis: ((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))) . b = (f . b) \ b
thus ((pair_diff A) .: (f,(incl (DISJOINT_PAIRS A)))) . b = (pair_diff A) . ((f . b),((incl (DISJOINT_PAIRS A)) . b)) by FUNCOP_1:48
.= (pair_diff A) . ((f . b),b) by FUNCT_1:35
.= (f . b) \ b by Def5 ; :: thesis: verum

end;

theorem Th25: :: HEYTING1:25

for A being set
for a being Element of DISJOINT_PAIRS A
for u, v being Element of (NormForm A) st ( for c being Element of DISJOINT_PAIRS A st c in u holds
ex b being Element of DISJOINT_PAIRS A st
( b in v & b c= c \/ a ) ) holds
ex b being Element of DISJOINT_PAIRS A st
( b in (@ u) =>> (@ v) & b c= a )

proof end;

Lm6: for A being set
for a being Element of DISJOINT_PAIRS A
for u being Element of (NormForm A)
for K being Element of Normal_forms_on A st a in K ^ (K =>> (@ u)) holds
ex b being Element of DISJOINT_PAIRS A st
( b in u & b c= a )

proof end;

theorem Th26: :: HEYTING1:26

for A being set
for K being Element of Normal_forms_on A holds K ^ (- K) = {}

proof end;

definition

let A be set ;
func pseudo_compl A -> UnOp of the carrier of (NormForm A) means :Def8: :: HEYTING1:def 8
for u being Element of (NormForm A) holds it . u = mi (- (@ u));
existence
ex b₁ being UnOp of the carrier of (NormForm A) st
for u being Element of (NormForm A) holds b₁ . u = mi (- (@ u))

proof end;

correctness
uniqueness
for b₁, b₂ being UnOp of the carrier of (NormForm A) st ( for u being Element of (NormForm A) holds b₁ . u = mi (- (@ u)) ) & ( for u being Element of (NormForm A) holds b₂ . u = mi (- (@ u)) ) holds
b₁ = b₂;

proof end;

func StrongImpl A -> BinOp of the carrier of (NormForm A) means :Def9: :: HEYTING1:def 9
for u, v being Element of (NormForm A) holds it . (u,v) = mi ((@ u) =>> (@ v));
existence
ex b₁ being BinOp of the carrier of (NormForm A) st
for u, v being Element of (NormForm A) holds b₁ . (u,v) = mi ((@ u) =>> (@ v))

proof end;

correctness
uniqueness
for b₁, b₂ being BinOp of the carrier of (NormForm A) st ( for u, v being Element of (NormForm A) holds b₁ . (u,v) = mi ((@ u) =>> (@ v)) ) & ( for u, v being Element of (NormForm A) holds b₂ . (u,v) = mi ((@ u) =>> (@ v)) ) holds
b₁ = b₂;

proof end;

let u be Element of (NormForm A);
func SUB u -> Element of Fin the carrier of (NormForm A) equals :: HEYTING1:def 10
bool u;
coherence
bool u is Element of Fin the carrier of (NormForm A)

proof end;

correctness
;
func diff u -> UnOp of the carrier of (NormForm A) means :Def11: :: HEYTING1:def 11
for v being Element of (NormForm A) holds it . v = u \ v;
existence
ex b₁ being UnOp of the carrier of (NormForm A) st
for v being Element of (NormForm A) holds b₁ . v = u \ v

proof end;

correctness
uniqueness
for b₁, b₂ being UnOp of the carrier of (NormForm A) st ( for v being Element of (NormForm A) holds b₁ . v = u \ v ) & ( for v being Element of (NormForm A) holds b₂ . v = u \ v ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def8 defines pseudo_compl HEYTING1:def 8 :
for A being set
for b₂ being UnOp of the carrier of (NormForm A) holds
( b₂ = pseudo_compl A iff for u being Element of (NormForm A) holds b₂ . u = mi (- (@ u)) );

:: deftheorem Def9 defines StrongImpl HEYTING1:def 9 :
for A being set
for b₂ being BinOp of the carrier of (NormForm A) holds
( b₂ = StrongImpl A iff for u, v being Element of (NormForm A) holds b₂ . (u,v) = mi ((@ u) =>> (@ v)) );

:: deftheorem defines SUB HEYTING1:def 10 :
for A being set
for u being Element of (NormForm A) holds SUB u = bool u;

:: deftheorem Def11 defines diff HEYTING1:def 11 :
for A being set
for u being Element of (NormForm A)
for b₃ being UnOp of the carrier of (NormForm A) holds
( b₃ = diff u iff for v being Element of (NormForm A) holds b₃ . v = u \ v );

deffunc H₁( set ) -> Element of bool [:[: the carrier of (NormForm $1), the carrier of (NormForm $1):], the carrier of (NormForm $1):] = the L_join of (NormForm $1);

deffunc H₂( set ) -> Element of bool [:[: the carrier of (NormForm $1), the carrier of (NormForm $1):], the carrier of (NormForm $1):] = the L_meet of (NormForm $1);

Lm7: for A being set
for u, v being Element of (NormForm A) st v in SUB u holds
v "\/" ((diff u) . v) = u

proof end;

theorem Th27: :: HEYTING1:27

for A being set
for u, v being Element of (NormForm A) holds (diff u) . v [= u

proof end;

Lm8: for A being set
for a being Element of DISJOINT_PAIRS A
for u being Element of (NormForm A) ex v being Element of (NormForm A) st
( v in SUB u & (@ v) ^ {a} = {} & ( for b being Element of DISJOINT_PAIRS A st b in (diff u) . v holds
b \/ a in DISJOINT_PAIRS A ) )

proof end;

theorem Th28: :: HEYTING1:28

for A being set
for u being Element of (NormForm A) holds u "/\" ((pseudo_compl A) . u) = Bottom (NormForm A)

proof end;

theorem Th29: :: HEYTING1:29

for A being set
for u, v being Element of (NormForm A) holds u "/\" ((StrongImpl A) . (u,v)) [= v

proof end;

theorem Th30: :: HEYTING1:30

for A being set
for a being Element of DISJOINT_PAIRS A
for u being Element of (NormForm A) st (@ u) ^ {a} = {} holds
(Atom A) . a [= (pseudo_compl A) . u

proof end;

theorem Th31: :: HEYTING1:31

for A being set
for a being Element of DISJOINT_PAIRS A
for u, w being Element of (NormForm A) st ( for b being Element of DISJOINT_PAIRS A st b in u holds
b \/ a in DISJOINT_PAIRS A ) & u "/\" ((Atom A) . a) [= w holds
(Atom A) . a [= (StrongImpl A) . (u,w)

proof end;

Lm9: now

let A be set ; :: thesis: for u, v being Element of (NormForm A) holds
( u "/\" H₃(u,v) [= v & ( for w being Element of (NormForm A) st u "/\" v [= w holds
v [= H₃(u,w) ) )
let u, v be Element of (NormForm A); :: thesis: ( u "/\" H₃(u,v) [= v & ( for w being Element of (NormForm A) st u "/\" v [= w holds
v [= H₃(u,w) ) )
deffunc H₃( Element of (NormForm A), Element of (NormForm A)) -> Element of the carrier of (NormForm A) = FinJoin ((SUB $1),(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff $1),$2)))));
set Psi = H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)));

A1: now

let w be Element of (NormForm A); :: thesis: ( w in SUB u implies (H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))) . w [= v )
set u2 = (diff u) . w;
set pc = (pseudo_compl A) . w;
set si = (StrongImpl A) . (((diff u) . w),v);
A2: w "/\" (((pseudo_compl A) . w) "/\" ((StrongImpl A) . (((diff u) . w),v))) = (w "/\" ((pseudo_compl A) . w)) "/\" ((StrongImpl A) . (((diff u) . w),v)) by LATTICES:def 7
.= (Bottom (NormForm A)) "/\" ((StrongImpl A) . (((diff u) . w),v)) by Th28
.= Bottom (NormForm A) by LATTICES:40 ;
assume w in SUB u ; :: thesis: (H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))) . w [= v
then A3: w "\/" ((diff u) . w) = u by Lm7;
(H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))) . w = H₂(A) . (u,((H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))) . w)) by FUNCOP_1:66
.= u "/\" ((H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))) . w) by LATTICES:def 2
.= u "/\" (H₂(A) . (((pseudo_compl A) . w),(((StrongImpl A) [:] ((diff u),v)) . w))) by FUNCOP_1:48
.= u "/\" (((pseudo_compl A) . w) "/\" (((StrongImpl A) [:] ((diff u),v)) . w)) by LATTICES:def 2
.= u "/\" (((pseudo_compl A) . w) "/\" ((StrongImpl A) . (((diff u) . w),v))) by FUNCOP_1:60
.= (w "/\" (((pseudo_compl A) . w) "/\" ((StrongImpl A) . (((diff u) . w),v)))) "\/" (((diff u) . w) "/\" (((pseudo_compl A) . w) "/\" ((StrongImpl A) . (((diff u) . w),v)))) by A3, LATTICES:def 11
.= ((diff u) . w) "/\" (((StrongImpl A) . (((diff u) . w),v)) "/\" ((pseudo_compl A) . w)) by A2, LATTICES:39
.= (((diff u) . w) "/\" ((StrongImpl A) . (((diff u) . w),v))) "/\" ((pseudo_compl A) . w) by LATTICES:def 7 ;
then A4: (H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))) . w [= ((diff u) . w) "/\" ((StrongImpl A) . (((diff u) . w),v)) by LATTICES:23;
((diff u) . w) "/\" ((StrongImpl A) . (((diff u) . w),v)) [= v by Th29;
hence (H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))) . w [= v by A4, LATTICES:25; :: thesis: verum

end;

u "/\" H₃(u,v) = FinJoin ((SUB u),(H₂(A) [;] (u,(H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v))))))) by LATTICE2:83;
hence u "/\" H₃(u,v) [= v by A1, LATTICE2:70; :: thesis: for w being Element of (NormForm A) st u "/\" v [= w holds
v [= H₃(u,w)
let w be Element of (NormForm A); :: thesis: ( u "/\" v [= w implies v [= H₃(u,w) )
assume A5: u "/\" v [= w ; :: thesis: v [= H₃(u,w)
A6: v = FinJoin ((@ v),(Atom A)) by Th15;
then A7: u "/\" v = FinJoin ((@ v),(H₂(A) [;] (u,(Atom A)))) by LATTICE2:83;

now

set pf = pseudo_compl A;
set sf = (StrongImpl A) [:] ((diff u),w);
let a be Element of DISJOINT_PAIRS A; :: thesis: ( a in @ v implies (Atom A) . a [= H₃(u,w) )
assume a in @ v ; :: thesis: (Atom A) . a [= H₃(u,w)
then (H₂(A) [;] (u,(Atom A))) . a [= w by A7, A5, LATTICE2:46;
then H₂(A) . (u,((Atom A) . a)) [= w by FUNCOP_1:66;
then A8: u "/\" ((Atom A) . a) [= w by LATTICES:def 2;
consider v being Element of (NormForm A) such that
A9: v in SUB u and
A10: (@ v) ^ {a} = {} and
A11: for b being Element of DISJOINT_PAIRS A st b in (diff u) . v holds
b \/ a in DISJOINT_PAIRS A by Lm8;
((diff u) . v) "/\" ((Atom A) . a) [= u "/\" ((Atom A) . a) by Th27, LATTICES:27;
then ((diff u) . v) "/\" ((Atom A) . a) [= w by A8, LATTICES:25;
then (Atom A) . a [= (StrongImpl A) . (((diff u) . v),w) by A11, Th31;
then A12: (Atom A) . a [= ((StrongImpl A) [:] ((diff u),w)) . v by FUNCOP_1:60;
A13: ((pseudo_compl A) . v) "/\" (((StrongImpl A) [:] ((diff u),w)) . v) = H₂(A) . (((pseudo_compl A) . v),(((StrongImpl A) [:] ((diff u),w)) . v)) by LATTICES:def 2
.= (H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),w)))) . v by FUNCOP_1:48 ;
(Atom A) . a [= (pseudo_compl A) . v by A10, Th30;
then (Atom A) . a [= (H₂(A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),w)))) . v by A12, A13, FILTER_0:7;
hence (Atom A) . a [= H₃(u,w) by A9, LATTICE2:44; :: thesis: verum

end;

hence v [= H₃(u,w) by A6, LATTICE2:70; :: thesis: verum

end;

Lm10: for A being set holds NormForm A is implicative

proof end;

registration

let A be set ;
cluster NormForm A -> implicative ;
coherence
NormForm A is implicative by Lm10;

end;

theorem :: HEYTING1:32

canceled;

theorem Th33: :: HEYTING1:33

for A being set
for u, v being Element of (NormForm A) holds u => v = FinJoin ((SUB u),( the L_meet of (NormForm A) .: ((pseudo_compl A),((StrongImpl A) [:] ((diff u),v)))))

proof end;

theorem :: HEYTING1:34

for A being set holds Top (NormForm A) = {[{},{}]}

proof end;