begin
definition
let A be ( ( ) ( )
set ) ;
assume
not
A : ( ( ) ( )
set ) is
empty
;
end;
registration
let A be ( ( ) ( )
set ) ;
end;
definition
let A be ( ( ) ( )
set ) ;
func Atom A -> ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
means
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) holds
it : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
= {a : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } : ( ( ) ( non
empty finite )
Element of
Normal_forms_on A : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ;
end;
theorem
for
A being ( ( ) ( )
set )
for
c,
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in (Atom A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) ) holds
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
= a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) holds
(Atom A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
= (singleton (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
bool [:(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,(Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
K being ( ( ) ( )
Element of
Normal_forms_on A : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) holds
FinJoin (
K : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ,
(Atom A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
= FinUnion (
K : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ,
(singleton (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
bool [:(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,(Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
definition
let A be ( ( ) ( )
set ) ;
func pair_diff A -> ( (
Function-like quasi_total ) ( non
empty V7()
V10(
[:[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
V11(
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
Function-like V17(
[:[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
quasi_total )
BinOp of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
means
for
a,
b being ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) holds
it : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
. (
a : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ,
b : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ) : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
= a : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
\ b : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) : ( ( ) ( )
Element of
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ;
end;
definition
let A be ( ( ) ( )
set ) ;
let B be ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
func - B -> ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
equals
(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
/\ { [ { (g : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) . t1 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) ( ) Element of [A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) where t1 is ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( g : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) . t1 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of [A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) in t1 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `2 : ( ( ) ( finite ) Element of Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) & t1 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) in B : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } , { (g : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) . t2 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) ( ) Element of [A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) where t2 is ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( g : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) . t2 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of [A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) in t2 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `1 : ( ( ) ( finite ) Element of Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) & t2 : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) in B : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } ] : ( ( ) ( V1() ) set ) where g is ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) ) : for s being ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) st s : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) in B : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) holds
g : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) . s : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of [A : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) in (s : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `1) : ( ( ) ( finite ) Element of Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) \/ (s : ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `2) : ( ( ) ( finite ) Element of Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) : ( ( ) ( finite ) Element of Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) } : ( ( ) (
V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
let C be ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
func B =>> C -> ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
equals
(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
/\ { (FinPairUnion (B : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,((pair_diff A : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10([:[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17([:[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) ) quasi_total ) BinOp of [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) .: (f : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) ,(incl (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of bool [:(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of bool [:(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) )) : ( ( ) ( ) Element of [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) where f is ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) : f : ( ( ) ( V7() V10( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) .: B : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) : ( ( ) ( finite ) Element of Fin [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) c= C : ( ( Function-like quasi_total ) ( V7() V10([:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ) V11(A : ( ( ) ( ) set ) ) Function-like quasi_total ) Element of bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) } : ( ( ) (
V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
end;
theorem
for
A being ( ( ) ( )
set )
for
B being ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
for
c being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in - B : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) holds
ex
g being ( ( ) (
V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
Funcs (
(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[A : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty functional )
FUNCTION_DOMAIN of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) ) ) st
( ( for
s being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
s : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in B : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) holds
g : ( ( ) (
V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
Funcs (
(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( non
empty functional )
FUNCTION_DOMAIN of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) ) )
. s : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of
[b1 : ( ( ) ( ) set ) ] : ( ( non
empty ) ( non
empty )
set ) )
in (s : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `1) : ( ( ) (
finite )
Element of
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
\/ (s : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `2) : ( ( ) (
finite )
Element of
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
finite )
Element of
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) &
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
= [ { (g : ( ( ) ( V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) ) . t1 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) ( ) Element of [b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) where t1 is ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( g : ( ( ) ( V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) ) . t1 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of [b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) in t1 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `2 : ( ( ) ( finite ) Element of Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) & t1 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) in B : ( ( ) ( finite ) Element of Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } , { (g : ( ( ) ( V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) ) . t2 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) ( ) Element of [b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) where t2 is ( ( ) ( ) Element of DISJOINT_PAIRS A : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( g : ( ( ) ( V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) ) . t2 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of [b1 : ( ( ) ( ) set ) ] : ( ( non empty ) ( non empty ) set ) ) in t2 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) `1 : ( ( ) ( finite ) Element of Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) & t2 : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) in B : ( ( ) ( finite ) Element of Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } ] : ( ( ) (
V1() )
set ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
B,
C being ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
for
c being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in B : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
=>> C : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) holds
ex
f being ( ( ) (
V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
Funcs (
(DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) : ( ( ) ( non
empty functional )
FUNCTION_DOMAIN of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ) st
(
f : ( ( ) (
V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
Funcs (
(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) : ( ( ) ( non
empty functional )
FUNCTION_DOMAIN of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) )
.: B : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
finite )
Element of
Fin [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
c= C : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) &
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
= FinPairUnion (
B : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
((pair_diff A : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10([:[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11([:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17([:[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) ) quasi_total ) BinOp of [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) .: (f : ( ( ) ( V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of Funcs ((DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) ) ,(incl (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11([:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Element of bool [:(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) )) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11(
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Element of
bool [:(DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ,[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
for
K being ( ( ) ( )
Element of
Normal_forms_on A : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
K : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
^ {a : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } : ( ( ) ( non
empty finite )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
= {} : ( ( ) (
empty V7()
non-empty empty-yielding finite finite-yielding V27() )
set ) holds
ex
b being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
(
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in - K : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) &
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
c= a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
for
u,
v being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st ( for
c being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
ex
b being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
(
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in v : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) &
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
c= c : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
\/ a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) ) ) ) holds
ex
b being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
(
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in (@ u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
=>> (@ v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) &
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
c= a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ) ;
definition
let A be ( ( ) ( )
set ) ;
func StrongImpl A -> ( (
Function-like quasi_total ) ( non
empty V7()
V10(
[: the carrier of (NormForm A : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm A : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
V11( the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
[: the carrier of (NormForm A : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm A : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
quasi_total )
BinOp of the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
means
for
u,
v being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
it : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
. (
u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
v : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
= mi ((@ u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of Normal_forms_on A : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) =>> (@ v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of Normal_forms_on A : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin A : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
Element of
Normal_forms_on A : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS A : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin A : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin A : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) ;
let u be ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ;
func diff u -> ( (
Function-like quasi_total ) ( non
empty V7()
V10( the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
V11( the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17( the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
quasi_total )
UnOp of the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
means
for
v being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
it : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
. v : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm A : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
= u : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
\ v : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) ) )
Element of
bool u : ( (
Function-like quasi_total ) (
V7()
V10(
[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) )
V11(
A : ( ( ) ( )
set ) )
Function-like quasi_total )
Element of
bool [:[:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( V7() ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) (
V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ;
end;
theorem
for
A being ( ( ) ( )
set )
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
for
u being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st
(@ u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
^ {a : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) } : ( ( ) ( non
empty finite )
Element of
Normal_forms_on b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty )
Element of
bool (Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) (
finite )
Element of
Fin (DISJOINT_PAIRS b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
= {} : ( ( ) (
empty V7()
non-empty empty-yielding finite finite-yielding V27() )
set ) holds
(Atom A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
[= (pseudo_compl A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
quasi_total )
UnOp of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
A being ( ( ) ( )
set )
for
a being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
for
u,
w being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st ( for
b being ( ( ) ( )
Element of
DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) st
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
in u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
b : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
\/ a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of
[:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
in DISJOINT_PAIRS A : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) &
u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) )
"/\" ((Atom A : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17( DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) quasi_total ) Function of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) . a : ( ( ) ( ) Element of DISJOINT_PAIRS b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V7() V10( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) V11( Fin b1 : ( ( ) ( ) set ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) Element of bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
[= w : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
(Atom A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
quasi_total )
Function of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. a : ( ( ) ( )
Element of
DISJOINT_PAIRS b1 : ( ( ) ( )
set ) : ( ( ) ( non
empty V7()
V10(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) )
V11(
Fin b1 : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) )
Element of
bool [:(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) ,(Fin b1 : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non empty cup-closed diff-closed preBoolean ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
[= (StrongImpl A : ( ( ) ( ) set ) ) : ( (
Function-like quasi_total ) ( non
empty V7()
V10(
[: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17(
[: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty V7() )
set ) )
quasi_total )
BinOp of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) )
. (
u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
w : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded )
LattStr ) : ( ( ) ( non
empty )
set ) ) ;
registration
let A be ( ( ) ( )
set ) ;
end;
theorem
for
A being ( ( ) ( )
set )
for
u,
v being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) holds
u : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) )
=> v : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) )
= FinJoin (
(SUB u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) (
finite )
Element of
Fin the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) : ( (
preBoolean ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ,
( the L_meet of (NormForm A : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( Function-like quasi_total ) ( non empty V7() V10([: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17([: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) ) quasi_total commutative associative idempotent ) Element of bool [:[: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) .: ((pseudo_compl A : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) UnOp of the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) ,((StrongImpl A : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10([: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17([: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) ) quasi_total ) BinOp of the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) [:] ((diff u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V7() V10( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) UnOp of the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V7() V10( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) V11( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) Function-like V17( the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) quasi_total ) Element of bool [: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V7() ) set ) : ( ( ) ( non empty cup-closed diff-closed preBoolean ) set ) ) )) : ( (
Function-like quasi_total ) ( non
empty V7()
V10( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) )
V11( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) )
Function-like V17( the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) )
quasi_total )
Element of
bool [: the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (NormForm b1 : ( ( ) ( ) set ) ) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non
empty V7() )
set ) : ( ( ) ( non
empty cup-closed diff-closed preBoolean )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
(NormForm b1 : ( ( ) ( ) set ) ) : ( (
strict ) ( non
empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded implicative Heyting )
LattStr ) : ( ( ) ( non
empty )
set ) ) ;