:: HEYTING1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

theorem Th1: :: HEYTING1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th3: :: HEYTING1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for B being Element of Fin (DISJOINT_PAIRS A) st B = {} holds
mi B = {}
proof end;

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;
assume that
A1: ( c in B & b in B ) and
c c= b ; :: thesis: c = b
( c = a & b = a ) by A1, TARSKI:def 1;
hence c = b ; :: 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;

Lm2: for A being set
for u, v being Element of (NormForm A) holds mi ((@ u) ^ (@ v)) = the L_meet of (NormForm A) . u,v
by NORMFORM:def 14;

Lm3: for A being set
for u, v being Element of (NormForm A) holds mi ((@ u) \/ (@ v)) = the L_join of (NormForm A) . u,v
by NORMFORM:def 14;

theorem :: HEYTING1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: HEYTING1:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: HEYTING1:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th7: :: HEYTING1:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for K being Element of Normal_forms_on A holds mi (K ^ K) = K
proof end;

theorem Th8: :: HEYTING1:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th10: :: HEYTING1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for a being Element of DISJOINT_PAIRS A holds a in (Atom A) . a
proof end;

theorem :: HEYTING1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for u being Element of (NormForm A) holds u = FinJoin (@ u),(Atom A)
proof end;

Lm4: 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;

Lm5: 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set holds [{} ,{} ] is Element of DISJOINT_PAIRS A
proof end;

theorem Th18: :: HEYTING1:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for K being Element of Normal_forms_on A st K = {} holds
- K = {[{} ,{} ]}
proof end;

theorem Th19: :: HEYTING1:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for a being Element of DISJOINT_PAIRS {} holds a = [{} ,{} ]
proof end;

theorem Th21: :: HEYTING1:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
DISJOINT_PAIRS {} = {[{} ,{} ]}
proof end;

Lm6: Fin (DISJOINT_PAIRS {} ) = {{} ,{[{} ,{} ]}}
proof end;

theorem Th22: :: HEYTING1:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set holds {[{} ,{} ]} is Element of Normal_forms_on A
proof end;

theorem Th23: :: HEYTING1:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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;

Lm7: 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for a being Element of DISJOINT_PAIRS A
for u, v 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 ) & ( 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;

Lm8: 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 ) -> M5([:the carrier of (NormForm $1),the carrier of (NormForm $1):],the carrier of (NormForm $1)) = the L_join of (NormForm $1);

deffunc H2( set ) -> M5([:the carrier of (NormForm $1),the carrier of (NormForm $1):],the carrier of (NormForm $1)) = the L_meet of (NormForm $1);

Lm9: 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set
for u, v being Element of (NormForm A) holds (diff u) . v [= u
proof end;

Lm10: 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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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;

Lm11: 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);
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;
assume w in SUB u ; :: thesis: (H2(A) [;] u,(H2(A) .: (pseudo_compl A),((StrongImpl A) [:] (diff u),v))) . w [= v
then A2: w "\/" ((diff u) . w) = u by Lm9;
A3: 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 ;
A4: ((diff u) . w) "/\" ((StrongImpl A) . ((diff u) . w),v) [= v by Th29;
(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 A2, LATTICES:def 11
.= ((diff u) . w) "/\" (((StrongImpl A) . ((diff u) . w),v) "/\" ((pseudo_compl A) . w)) by A3, LATTICES:39
.= (((diff u) . w) "/\" ((StrongImpl A) . ((diff u) . w),v)) "/\" ((pseudo_compl A) . w) by LATTICES:def 7 ;
then (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;
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) )
A5: v = FinJoin (@ v),(Atom A) by Th15;
then A6: u "/\" v = FinJoin (@ v),(H2(A) [;] u,(Atom A)) by LATTICE2:83;
assume A7: u "/\" v [= w ; :: thesis: v [= H3(u,w)
now
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 A6, A7, 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 Lm10;
(diff u) . v [= u by Th27;
then ((diff u) . v) "/\" ((Atom A) . a) [= u "/\" ((Atom A) . a) by LATTICES:27;
then A12: ((diff u) . v) "/\" ((Atom A) . a) [= w by A8, LATTICES:25;
set pf = pseudo_compl A;
set sf = (StrongImpl A) [:] (diff u),w;
A13: (Atom A) . a [= (pseudo_compl A) . v by A10, Th30;
(Atom A) . a [= (StrongImpl A) . ((diff u) . v),w by A11, A12, Th31;
then A14: (Atom A) . a [= ((StrongImpl A) [:] (diff u),w) . v by FUNCOP_1:60;
((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 ;
then (Atom A) . a [= (H2(A) .: (pseudo_compl A),((StrongImpl A) [:] (diff u),w)) . v by A13, A14, FILTER_0:7;
hence (Atom A) . a [= H3(u,w) by A9, LATTICE2:44; :: thesis: verum
end;
hence v [= H3(u,w) by A5, LATTICE2:70; :: thesis: verum
end;

Lm12: 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 Lm12;
end;

theorem :: HEYTING1:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th33: :: HEYTING1:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being set holds Top (NormForm A) = {[{} ,{} ]}
proof end;