:: Lattice of Substitutions Is a Heyting Algebra
::
:: Copyright (c) 1998-2021 Association of Mizar Users

theorem Th1: :: HEYTING2:1
for V, C, a, b being set st b in SubstitutionSet (V,C) & a in b holds
a is finite Function
proof end;

Lm1: for A, B, C being set st A = B \/ C & A c= B holds
A = B

by XBOOLE_1:7;

theorem Th2: :: HEYTING2:2
for V, C being set
for a being finite Element of PFuncs (V,C) holds {a} in SubstitutionSet (V,C)
proof end;

theorem :: HEYTING2:3
for V, C being set
for A, B being Element of SubstitutionSet (V,C) st A ^ B = A holds
for a being set st a in A holds
ex b being set st
( b in B & b c= a )
proof end;

theorem Th4: :: HEYTING2:4
for V, C being set
for A, B being Element of SubstitutionSet (V,C) st mi (A ^ B) = A holds
for a being set st a in A holds
ex b being set st
( b in B & b c= a )
proof end;

theorem Th5: :: HEYTING2:5
for V, C being set
for A, B being Element of SubstitutionSet (V,C) st ( for a being set st a in A holds
ex b being set st
( b in B & b c= a ) ) holds
mi (A ^ B) = A
proof end;

definition
let V be set ;
let C be finite set ;
let A be Element of Fin (PFuncs (V,C));
func Involved A -> set means :Def1: :: HEYTING2:def 1
for x being object holds
( x in it iff ex f being finite Function st
( f in A & x in dom f ) );
existence
ex b1 being set st
for x being object holds
( x in b1 iff ex f being finite Function st
( f in A & x in dom f ) )
proof end;
uniqueness
for b1, b2 being set st ( for x being object holds
( x in b1 iff ex f being finite Function st
( f in A & x in dom f ) ) ) & ( for x being object holds
( x in b2 iff ex f being finite Function st
( f in A & x in dom f ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Involved HEYTING2:def 1 :
for V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C))
for b4 being set holds
( b4 = Involved A iff for x being object holds
( x in b4 iff ex f being finite Function st
( f in A & x in dom f ) ) );

theorem Th6: :: HEYTING2:6
for V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C)) holds Involved A c= V
proof end;

Lm2: for V being set
for C being finite set
for A being non empty Element of Fin (PFuncs (V,C)) holds Involved A is finite

proof end;

theorem Th7: :: HEYTING2:7
for V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C)) st A = {} holds
Involved A = {}
proof end;

registration
let V be set ;
let C be finite set ;
let A be Element of Fin (PFuncs (V,C));
coherence
proof end;
end;

theorem :: HEYTING2:8
for C being finite set
for A being Element of Fin (PFuncs ({},C)) holds Involved A = {} by ;

definition
let V be set ;
let C be finite set ;
let A be Element of Fin (PFuncs (V,C));
func - A -> Element of Fin (PFuncs (V,C)) equals :: HEYTING2:def 2
{ f where f is Element of PFuncs ((),C) : for g being Element of PFuncs (V,C) st g in A holds
not f tolerates g
}
;
coherence
{ f where f is Element of PFuncs ((),C) : for g being Element of PFuncs (V,C) st g in A holds
not f tolerates g
}
is Element of Fin (PFuncs (V,C))
proof end;
end;

:: deftheorem defines - HEYTING2:def 2 :
for V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C)) holds - A = { f where f is Element of PFuncs ((),C) : for g being Element of PFuncs (V,C) st g in A holds
not f tolerates g
}
;

theorem Th9: :: HEYTING2:9
for V being set
for C being finite set
for A being Element of SubstitutionSet (V,C) holds A ^ (- A) = {}
proof end;

theorem Th10: :: HEYTING2:10
for V being set
for C being finite set
for A being Element of SubstitutionSet (V,C) st A = {} holds
- A =
proof end;

theorem :: HEYTING2:11
for V being set
for C being finite set
for A being Element of SubstitutionSet (V,C) st A = holds
- A = {}
proof end;

theorem Th12: :: HEYTING2:12
for V being set
for C being finite set
for A being Element of SubstitutionSet (V,C) holds mi (A ^ (- A)) = Bottom (SubstLatt (V,C))
proof end;

theorem :: HEYTING2:13
for V being non empty set
for C being non empty finite set
for A being Element of SubstitutionSet (V,C) st A = {} holds
mi (- A) = Top (SubstLatt (V,C))
proof end;

theorem Th14: :: HEYTING2:14
for V being set
for C being finite set
for A being Element of SubstitutionSet (V,C)
for a being Element of PFuncs (V,C)
for B being Element of SubstitutionSet (V,C) st B = {a} & A ^ B = {} holds
ex b being finite set st
( b in - A & b c= a )
proof end;

definition
let V be set ;
let C be finite set ;
let A, B be Element of Fin (PFuncs (V,C));
func A =>> B -> Element of Fin (PFuncs (V,C)) equals :: HEYTING2:def 3
(PFuncs (V,C)) /\ { (union { ((f . i) \ i) where i is Element of PFuncs (V,C) : i in A } ) where f is Element of PFuncs (A,B) : dom f = A } ;
coherence
(PFuncs (V,C)) /\ { (union { ((f . i) \ i) where i is Element of PFuncs (V,C) : i in A } ) where f is Element of PFuncs (A,B) : dom f = A } is Element of Fin (PFuncs (V,C))
proof end;
end;

:: deftheorem defines =>> HEYTING2:def 3 :
for V being set
for C being finite set
for A, B being Element of Fin (PFuncs (V,C)) holds A =>> B = (PFuncs (V,C)) /\ { (union { ((f . i) \ i) where i is Element of PFuncs (V,C) : i in A } ) where f is Element of PFuncs (A,B) : dom f = A } ;

theorem Th15: :: HEYTING2:15
for V being set
for C being finite set
for A, B being Element of Fin (PFuncs (V,C))
for s being set st s in A =>> B holds
ex f being PartFunc of A,B st
( s = union { ((f . i) \ i) where i is Element of PFuncs (V,C) : i in A } & dom f = A )
proof end;

Lm3: for a, V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C))
for K being Element of SubstitutionSet (V,C) st a in A ^ (A =>> K) holds
ex b being set st
( b in K & b c= a )

proof end;

theorem :: HEYTING2:16
for V being set
for C being finite set
for A being Element of Fin (PFuncs (V,C)) st A = {} holds
A =>> A =
proof end;

Lm4: for V being set
for C being finite set
for K being Element of SubstitutionSet (V,C)
for X being set st X c= K holds
X in SubstitutionSet (V,C)

proof end;

theorem Th17: :: HEYTING2:17
for V being set
for C being finite set
for u being Element of (SubstLatt (V,C))
for X being set st X c= u holds
X is Element of (SubstLatt (V,C))
proof end;

definition
let V be set ;
let C be finite set ;
func pseudo_compl (V,C) -> UnOp of the carrier of (SubstLatt (V,C)) means :Def4: :: HEYTING2:def 4
for u being Element of (SubstLatt (V,C))
for u9 being Element of SubstitutionSet (V,C) st u9 = u holds
it . u = mi (- u9);
existence
ex b1 being UnOp of the carrier of (SubstLatt (V,C)) st
for u being Element of (SubstLatt (V,C))
for u9 being Element of SubstitutionSet (V,C) st u9 = u holds
b1 . u = mi (- u9)
proof end;
correctness
uniqueness
for b1, b2 being UnOp of the carrier of (SubstLatt (V,C)) st ( for u being Element of (SubstLatt (V,C))
for u9 being Element of SubstitutionSet (V,C) st u9 = u holds
b1 . u = mi (- u9) ) & ( for u being Element of (SubstLatt (V,C))
for u9 being Element of SubstitutionSet (V,C) st u9 = u holds
b2 . u = mi (- u9) ) holds
b1 = b2
;
proof end;
func StrongImpl (V,C) -> BinOp of the carrier of (SubstLatt (V,C)) means :Def5: :: HEYTING2:def 5
for u, v being Element of (SubstLatt (V,C))
for u9, v9 being Element of SubstitutionSet (V,C) st u9 = u & v9 = v holds
it . (u,v) = mi (u9 =>> v9);
existence
ex b1 being BinOp of the carrier of (SubstLatt (V,C)) st
for u, v being Element of (SubstLatt (V,C))
for u9, v9 being Element of SubstitutionSet (V,C) st u9 = u & v9 = v holds
b1 . (u,v) = mi (u9 =>> v9)
proof end;
correctness
uniqueness
for b1, b2 being BinOp of the carrier of (SubstLatt (V,C)) st ( for u, v being Element of (SubstLatt (V,C))
for u9, v9 being Element of SubstitutionSet (V,C) st u9 = u & v9 = v holds
b1 . (u,v) = mi (u9 =>> v9) ) & ( for u, v being Element of (SubstLatt (V,C))
for u9, v9 being Element of SubstitutionSet (V,C) st u9 = u & v9 = v holds
b2 . (u,v) = mi (u9 =>> v9) ) holds
b1 = b2
;
proof end;
let u be Element of (SubstLatt (V,C));
func SUB u -> Element of Fin the carrier of (SubstLatt (V,C)) equals :: HEYTING2:def 6
bool u;
coherence
bool u is Element of Fin the carrier of (SubstLatt (V,C))
proof end;
correctness
;
func diff u -> UnOp of the carrier of (SubstLatt (V,C)) means :Def7: :: HEYTING2:def 7
for v being Element of (SubstLatt (V,C)) holds it . v = u \ v;
existence
ex b1 being UnOp of the carrier of (SubstLatt (V,C)) st
for v being Element of (SubstLatt (V,C)) holds b1 . v = u \ v
proof end;
correctness
uniqueness
for b1, b2 being UnOp of the carrier of (SubstLatt (V,C)) st ( for v being Element of (SubstLatt (V,C)) holds b1 . v = u \ v ) & ( for v being Element of (SubstLatt (V,C)) holds b2 . v = u \ v ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def4 defines pseudo_compl HEYTING2:def 4 :
for V being set
for C being finite set
for b3 being UnOp of the carrier of (SubstLatt (V,C)) holds
( b3 = pseudo_compl (V,C) iff for u being Element of (SubstLatt (V,C))
for u9 being Element of SubstitutionSet (V,C) st u9 = u holds
b3 . u = mi (- u9) );

:: deftheorem Def5 defines StrongImpl HEYTING2:def 5 :
for V being set
for C being finite set
for b3 being BinOp of the carrier of (SubstLatt (V,C)) holds
( b3 = StrongImpl (V,C) iff for u, v being Element of (SubstLatt (V,C))
for u9, v9 being Element of SubstitutionSet (V,C) st u9 = u & v9 = v holds
b3 . (u,v) = mi (u9 =>> v9) );

:: deftheorem defines SUB HEYTING2:def 6 :
for V being set
for C being finite set
for u being Element of (SubstLatt (V,C)) holds SUB u = bool u;

:: deftheorem Def7 defines diff HEYTING2:def 7 :
for V being set
for C being finite set
for u being Element of (SubstLatt (V,C))
for b4 being UnOp of the carrier of (SubstLatt (V,C)) holds
( b4 = diff u iff for v being Element of (SubstLatt (V,C)) holds b4 . v = u \ v );

Lm5: for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) st v in SUB u holds
v "\/" ((diff u) . v) = u

proof end;

Lm6: for V being set
for C being finite set
for u being Element of (SubstLatt (V,C))
for a being Element of PFuncs (V,C)
for K being Element of Fin (PFuncs (V,C)) st K = {a} holds
ex v being Element of SubstitutionSet (V,C) st
( v in SUB u & v ^ K = {} & ( for b being Element of PFuncs (V,C) st b in (diff u) . v holds
b tolerates a ) )

proof end;

definition
let V be set ;
let C be finite set ;
func Atom (V,C) -> Function of (PFuncs (V,C)), the carrier of (SubstLatt (V,C)) means :Def8: :: HEYTING2:def 8
for a being Element of PFuncs (V,C) holds it . a = mi {.a.};
existence
ex b1 being Function of (PFuncs (V,C)), the carrier of (SubstLatt (V,C)) st
for a being Element of PFuncs (V,C) holds b1 . a = mi {.a.}
proof end;
correctness
uniqueness
for b1, b2 being Function of (PFuncs (V,C)), the carrier of (SubstLatt (V,C)) st ( for a being Element of PFuncs (V,C) holds b1 . a = mi {.a.} ) & ( for a being Element of PFuncs (V,C) holds b2 . a = mi {.a.} ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def8 defines Atom HEYTING2:def 8 :
for V being set
for C being finite set
for b3 being Function of (PFuncs (V,C)), the carrier of (SubstLatt (V,C)) holds
( b3 = Atom (V,C) iff for a being Element of PFuncs (V,C) holds b3 . a = mi {.a.} );

Lm7: for V being set
for C being finite set
for a being Element of PFuncs (V,C) st a is finite holds
(Atom (V,C)) . a = {a}

proof end;

theorem Th18: :: HEYTING2:18
for V being set
for C being finite set
for K being Element of SubstitutionSet (V,C) holds FinJoin (K,(Atom (V,C))) = FinUnion (K,(singleton (PFuncs (V,C))))
proof end;

theorem Th19: :: HEYTING2:19
for V being set
for C being finite set
for u being Element of SubstitutionSet (V,C) holds u = FinJoin (u,(Atom (V,C)))
proof end;

Lm8: for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) st ( for a being set st a in u holds
ex b being set st
( b in v & b c= a ) ) holds
u [= v

proof end;

theorem Th20: :: HEYTING2:20
for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) holds (diff u) . v [= u
proof end;

theorem Th21: :: HEYTING2:21
for V being set
for C being finite set
for a being Element of PFuncs (V,C) st a is finite holds
for c being set st c in (Atom (V,C)) . a holds
c = a
proof end;

theorem Th22: :: HEYTING2:22
for V being set
for C being finite set
for u being Element of (SubstLatt (V,C))
for K, L being Element of SubstitutionSet (V,C)
for a being Element of PFuncs (V,C) st K = {a} & L = u & L ^ K = {} holds
(Atom (V,C)) . a [= (pseudo_compl (V,C)) . u
proof end;

theorem Th23: :: HEYTING2:23
for V being set
for C being finite set
for a being finite Element of PFuncs (V,C) holds a in (Atom (V,C)) . a
proof end;

Lm9: for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) st u [= v holds
for x being set st x in u holds
ex b being set st
( b in v & b c= x )

proof end;

theorem Th24: :: HEYTING2:24
for V being set
for C being finite set
for a being Element of PFuncs (V,C)
for u, v being Element of SubstitutionSet (V,C) st ( for c being set st c in u holds
ex b being set st
( b in v & b c= c \/ a ) ) holds
ex b being set st
( b in u =>> v & b c= a )
proof end;

theorem Th25: :: HEYTING2:25
for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C))
for a being finite Element of PFuncs (V,C) st ( for b being Element of PFuncs (V,C) st b in u holds
b tolerates a ) & u "/\" ((Atom (V,C)) . a) [= v holds
(Atom (V,C)) . a [= (StrongImpl (V,C)) . (u,v)
proof end;

deffunc H1( set , set ) -> Element of bool [:[: the carrier of (SubstLatt ($1,$2)), the carrier of (SubstLatt ($1,$2)):], the carrier of (SubstLatt ($1,$2)):] = the L_meet of (SubstLatt ($1,$2));

theorem Th26: :: HEYTING2:26
for V being set
for C being finite set
for u being Element of (SubstLatt (V,C)) holds u "/\" ((pseudo_compl (V,C)) . u) = Bottom (SubstLatt (V,C))
proof end;

theorem Th27: :: HEYTING2:27
for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) holds u "/\" ((StrongImpl (V,C)) . (u,v)) [= v
proof end;

Lm10: now :: thesis: for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) holds
( u "/\" H2(u,v) [= v & ( for w being Element of (SubstLatt (V,C)) st u "/\" v [= w holds
v [= H2(u,w) ) )
let V be set ; :: thesis: for C being finite set
for u, v being Element of (SubstLatt (V,C)) holds
( u "/\" H2(u,v) [= v & ( for w being Element of (SubstLatt (V,C)) st u "/\" v [= w holds
v [= H2(u,w) ) )

let C be finite set ; :: thesis: for u, v being Element of (SubstLatt (V,C)) holds
( u "/\" H2(u,v) [= v & ( for w being Element of (SubstLatt (V,C)) st u "/\" v [= w holds
v [= H2(u,w) ) )

let u, v be Element of (SubstLatt (V,C)); :: thesis: ( u "/\" H2(u,v) [= v & ( for w being Element of (SubstLatt (V,C)) st u "/\" v [= w holds
v [= H2(u,w) ) )

deffunc H2( Element of (SubstLatt (V,C)), Element of (SubstLatt (V,C))) -> Element of the carrier of (SubstLatt (V,C)) = FinJoin ((SUB $1),(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff$1),\$2)))));
set Psi = H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)));
reconsider v9 = v as Element of SubstitutionSet (V,C) by SUBSTLAT:def 4;
A1: now :: thesis: for w being Element of (SubstLatt (V,C)) st w in SUB u holds
(H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w [= v
let w be Element of (SubstLatt (V,C)); :: thesis: ( w in SUB u implies (H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w [= v )
set u2 = (diff u) . w;
set pc = (pseudo_compl (V,C)) . w;
set si = (StrongImpl (V,C)) . (((diff u) . w),v);
A2: w "/\" (((pseudo_compl (V,C)) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v))) = (w "/\" ((pseudo_compl (V,C)) . w)) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)) by LATTICES:def 7
.= (Bottom (SubstLatt (V,C))) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)) by Th26
.= Bottom (SubstLatt (V,C)) ;
assume w in SUB u ; :: thesis: (H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w [= v
then A3: w "\/" ((diff u) . w) = u by Lm5;
(H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w = H1(V,C) . (u,((H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))) . w)) by FUNCOP_1:53
.= u "/\" ((H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))) . w) by LATTICES:def 2
.= u "/\" (H1(V,C) . (((pseudo_compl (V,C)) . w),(((StrongImpl (V,C)) [:] ((diff u),v)) . w))) by FUNCOP_1:37
.= u "/\" (((pseudo_compl (V,C)) . w) "/\" (((StrongImpl (V,C)) [:] ((diff u),v)) . w)) by LATTICES:def 2
.= u "/\" (((pseudo_compl (V,C)) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v))) by FUNCOP_1:48
.= (w "/\" (((pseudo_compl (V,C)) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)))) "\/" (((diff u) . w) "/\" (((pseudo_compl (V,C)) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)))) by
.= ((diff u) . w) "/\" (((StrongImpl (V,C)) . (((diff u) . w),v)) "/\" ((pseudo_compl (V,C)) . w)) by A2
.= (((diff u) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v))) "/\" ((pseudo_compl (V,C)) . w) by LATTICES:def 7 ;
then A4: (H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w [= ((diff u) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)) by LATTICES:6;
((diff u) . w) "/\" ((StrongImpl (V,C)) . (((diff u) . w),v)) [= v by Th27;
hence (H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))) . w [= v by ; :: thesis: verum
end;
u "/\" H2(u,v) = FinJoin ((SUB u),(H1(V,C) [;] (u,(H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v))))))) by LATTICE2:66;
hence u "/\" H2(u,v) [= v by ; :: thesis: for w being Element of (SubstLatt (V,C)) st u "/\" v [= w holds
v [= H2(u,w)

let w be Element of (SubstLatt (V,C)); :: thesis: ( u "/\" v [= w implies v [= H2(u,w) )
assume A5: u "/\" v [= w ; :: thesis: v [= H2(u,w)
A6: v = FinJoin (v9,(Atom (V,C))) by Th19;
then A7: u "/\" v = FinJoin (v9,(H1(V,C) [;] (u,(Atom (V,C))))) by LATTICE2:66;
now :: thesis: for a being Element of PFuncs (V,C) st a in v9 holds
(Atom (V,C)) . a [= H2(u,w)
set pf = pseudo_compl (V,C);
set sf = (StrongImpl (V,C)) [:] ((diff u),w);
let a be Element of PFuncs (V,C); :: thesis: ( a in v9 implies (Atom (V,C)) . a [= H2(u,w) )
reconsider SA = {.a.} as Element of Fin (PFuncs (V,C)) ;
consider v being Element of SubstitutionSet (V,C) such that
A8: v in SUB u and
A9: v ^ SA = {} and
A10: for b being Element of PFuncs (V,C) st b in (diff u) . v holds
b tolerates a by Lm6;
assume A11: a in v9 ; :: thesis: (Atom (V,C)) . a [= H2(u,w)
then A12: a is finite by Th1;
reconsider v9 = v as Element of (SubstLatt (V,C)) by SUBSTLAT:def 4;
set dv = (diff u) . v9;
(H1(V,C) [;] (u,(Atom (V,C)))) . a [= w by ;
then H1(V,C) . (u,((Atom (V,C)) . a)) [= w by FUNCOP_1:53;
then A13: u "/\" ((Atom (V,C)) . a) [= w by LATTICES:def 2;
a is finite by ;
then reconsider SS = {a} as Element of SubstitutionSet (V,C) by Th2;
v ^ SS = {} by A9;
then A14: (Atom (V,C)) . a [= (pseudo_compl (V,C)) . v9 by Th22;
((diff u) . v9) "/\" ((Atom (V,C)) . a) [= u "/\" ((Atom (V,C)) . a) by ;
then ((diff u) . v9) "/\" ((Atom (V,C)) . a) [= w by ;
then (Atom (V,C)) . a [= (StrongImpl (V,C)) . (((diff u) . v9),w) by ;
then A15: (Atom (V,C)) . a [= ((StrongImpl (V,C)) [:] ((diff u),w)) . v9 by FUNCOP_1:48;
((pseudo_compl (V,C)) . v9) "/\" (((StrongImpl (V,C)) [:] ((diff u),w)) . v9) = H1(V,C) . (((pseudo_compl (V,C)) . v9),(((StrongImpl (V,C)) [:] ((diff u),w)) . v9)) by LATTICES:def 2
.= (H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),w)))) . v9 by FUNCOP_1:37 ;
then (Atom (V,C)) . a [= (H1(V,C) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),w)))) . v9 by ;
hence (Atom (V,C)) . a [= H2(u,w) by ; :: thesis: verum
end;
hence v [= H2(u,w) by ; :: thesis: verum
end;

Lm11: for V being set
for C being finite set holds SubstLatt (V,C) is implicative

proof end;

registration
let V be set ;
let C be finite set ;
cluster SubstLatt (V,C) -> implicative ;
coherence
SubstLatt (V,C) is implicative
by Lm11;
end;

theorem :: HEYTING2:28
for V being set
for C being finite set
for u, v being Element of (SubstLatt (V,C)) holds u => v = FinJoin ((SUB u),( the L_meet of (SubstLatt (V,C)) .: ((pseudo_compl (V,C)),((StrongImpl (V,C)) [:] ((diff u),v)))))
proof end;