:: CLOSURE2 semantic presentation
begin
notation
let
I
be ( ( ) ( )
set
) ;
let
A
,
B
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
synonym
A
in'
B
for
A
in
B
;
end;
notation
let
I
be ( ( ) ( )
set
) ;
let
A
,
B
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
synonym
A
c='
B
for
A
c=
B
;
end;
theorem
:: CLOSURE2:1
for
M
being ( ( non
empty
) ( non
empty
)
set
)
for
X
,
Y
being ( ( ) ( )
Element
of
M
: ( ( non
empty
) ( non
empty
)
set
) ) st
X
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
c=
Y
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) holds
(
id
M
: ( ( non
empty
) ( non
empty
)
set
)
)
: ( (
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-valued
Function-like
one-to-one
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
quasi_total
)
Element
of
bool
[:
b
1
: ( ( non
empty
) ( non
empty
)
set
) ,
b
1
: ( ( non
empty
) ( non
empty
)
set
)
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
.
X
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
c=
(
id
M
: ( ( non
empty
) ( non
empty
)
set
)
)
: ( (
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-valued
Function-like
one-to-one
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
quasi_total
)
Element
of
bool
[:
b
1
: ( ( non
empty
) ( non
empty
)
set
) ,
b
1
: ( ( non
empty
) ( non
empty
)
set
)
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
.
Y
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: CLOSURE2:2
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
B
being ( ( ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSubset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) holds
rng
B
: ( ( ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSubset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
)
set
)
c=
union
(
rng
(
bool
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
ManySortedSubset
of
bool
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ( non
empty
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V17
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ;
begin
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
func
Bool
M
->
( ( ) ( )
set
)
means
:: CLOSURE2:def 1
for
x
being ( ( ) ( )
set
) holds
(
x
: ( ( ) ( )
set
)
in
it
: ( ( ) (
Relation-like
the
carrier'
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier'
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) )
ManySortedFunction
of the
Arity
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ,
K180
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ))
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
K183
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ,
M
: ( ( ) ( )
MSAlgebra
over
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) ) ) : ( (
Relation-like
K180
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) )) ) ) (
Relation-like
K180
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) )) ) )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
) , the
ResultSort
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
) , the
carrier
of
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) : ( ( ) ( )
set
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
M
: ( ( ) ( )
MSAlgebra
over
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) ) : ( (
Relation-like
) (
Relation-like
)
set
) ) iff
x
: ( ( ) ( )
set
) is ( ( ) (
Relation-like
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
)
-defined
Function-like
V17
(
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) ) )
ManySortedSubset
of
M
: ( ( ) ( )
MSAlgebra
over
I
: ( ( non
empty
) ( non
empty
)
ManySortedSign
) ) ) );
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( ( ) ( )
set
)
->
non
empty
functional
with_common_domain
;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
mode
SubsetFamily of
M
is
( ( ) (
functional
)
Subset
of ( ( ) ( non
empty
)
set
) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
:: original:
Bool
redefine
func
Bool
M
->
( ( ) (
functional
)
SubsetFamily
of
M
: ( ( ) ( )
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
non
empty
functional
with_common_domain
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
empty
functional
finite
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
S
be ( ( non
empty
) ( non
empty
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
:: original:
Element
redefine
mode
Element
of
S
->
( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ;
end;
theorem
:: CLOSURE2:3
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
,
SG
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
\/
SG
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( ( ) (
functional
)
Element
of
bool
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:4
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
,
SG
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
/\
SG
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( ( ) (
functional
)
Element
of
bool
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:5
for
I
,
x
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
\
x
: ( ( ) ( )
set
) : ( ( ) (
functional
)
Element
of
bool
(
Bool
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:6
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
,
SG
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
\+\
SG
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( ( ) (
functional
)
Element
of
bool
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:7
for
I
being ( ( ) ( )
set
)
for
A
,
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) st
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
c=
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) holds
{
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
}
: ( ( ) ( non
empty
functional
finite
)
set
) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:8
for
I
being ( ( ) ( )
set
)
for
A
,
M
,
B
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) st
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
c=
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) &
B
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
c=
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) holds
{
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ,
B
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
}
: ( ( ) ( non
empty
functional
finite
)
set
) is ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:9
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) holds
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
/\
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
in
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:10
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) holds
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
\/
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
in
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:11
for
I
being ( ( ) ( )
set
)
for
A
,
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) holds
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
\
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
in
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:12
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) holds
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
\+\
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
in
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
begin
definition
let
S
be ( (
functional
) (
functional
)
set
) ;
func
|.
S
.|
->
( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
means
:: CLOSURE2:def 2
ex
A
being ( ( non
empty
functional
) ( non
empty
functional
)
set
) st
(
A
: ( ( non
empty
functional
) ( non
empty
functional
)
set
)
=
S
: ( ( ) ( )
set
) &
dom
it
: ( (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) ) (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) )
set
) : ( ( ) ( )
set
)
=
meet
{
(
dom
x
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) where
x
is ( ( ) (
Relation-like
Function-like
)
Element
of
A
: ( ( non
empty
functional
) ( non
empty
functional
)
set
) ) : verum
}
: ( ( ) ( )
set
) & ( for
i
being ( ( ) ( )
set
) st
i
: ( ( ) ( )
set
)
in
dom
it
: ( (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) ) (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) )
set
) : ( ( ) ( )
set
) holds
it
: ( (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) ) (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) )
set
)
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
x
: ( ( ) (
Relation-like
Function-like
)
Element
of
b
1
: ( ( non
empty
functional
) ( non
empty
functional
)
set
) )
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) where
x
is ( ( ) (
Relation-like
Function-like
)
Element
of
A
: ( ( non
empty
functional
) ( non
empty
functional
)
set
) ) : verum
}
) )
if
S
: ( ( ) ( )
set
)
<>
{}
: ( ( ) (
empty
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
set
)
otherwise
it
: ( (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) ) (
Relation-like
S
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
set
) ) )
set
)
=
{}
: ( ( ) (
empty
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
set
) ;
end;
theorem
:: CLOSURE2:13
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( non
empty
) ( non
empty
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
dom
|.
SF
: ( ( non
empty
) ( non
empty
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
)
=
I
: ( ( ) ( )
set
) ;
registration
let
S
be ( (
empty
functional
) (
empty
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
set
) ;
cluster
|.
S
: ( (
empty
functional
) (
empty
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
->
empty
Relation-like
Function-like
;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
S
be ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
func
|:
S
:|
->
( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
equals
:: CLOSURE2:def 3
|.
S
: ( ( ) (
Relation-like
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
ManySortedFunction
of the
Arity
of
I
: ( ( ) ( )
set
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ,
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
K183
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ,
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) : ( (
Relation-like
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )) ) ) (
Relation-like
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )) ) )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
) , the
ResultSort
of
I
: ( ( ) ( )
set
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) , the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
) )
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
if
S
: ( ( ) (
Relation-like
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
ManySortedFunction
of the
Arity
of
I
: ( ( ) ( )
set
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ,
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
K183
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ,
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) : ( (
Relation-like
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )) ) ) (
Relation-like
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ))
-defined
Function-like
V17
(
K180
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
M12
( the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )) ) )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
) , the
ResultSort
of
I
: ( ( ) ( )
set
) : ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
the
carrier'
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
) , the
carrier
of
I
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
*
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
) )
<>
{}
: ( ( ) (
empty
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
set
)
otherwise
[[0]]
I
: ( ( ) ( )
set
) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
empty-yielding
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
S
be ( (
empty
) (
empty
proper
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
cluster
|:
S
: ( (
empty
) (
empty
proper
Relation-like
non-empty
empty-yielding
Function-like
one-to-one
constant
functional
Function-yielding
V25
()
finite
finite-yielding
V45
() )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
:|
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
->
Relation-like
V9
()
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ;
end;
theorem
:: CLOSURE2:14
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st not
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
empty
holds
for
i
being ( ( ) ( )
set
) st
i
: ( ( ) ( )
set
)
in
I
: ( ( ) ( )
set
) holds
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
x
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) where
x
is ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) :
x
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
in
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
}
;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
SF
be ( ( non
empty
) ( non
empty
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
cluster
|:
SF
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
:|
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
->
Relation-like
V8
()
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ;
end;
theorem
:: CLOSURE2:15
for
f
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) holds
dom
|.
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
)
=
dom
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
) ;
theorem
:: CLOSURE2:16
for
f
,
f1
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) holds
dom
|.
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) ,
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
)
=
(
dom
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
)
: ( ( ) ( )
set
)
/\
(
dom
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ;
theorem
:: CLOSURE2:17
for
i
being ( ( ) ( )
set
)
for
f
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) st
i
: ( ( ) ( )
set
)
in
dom
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
) holds
|.
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
finite
)
set
) ;
theorem
:: CLOSURE2:18
for
i
,
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
f
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) ) st
i
: ( ( ) ( )
set
)
in
I
: ( ( ) ( )
set
) &
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) )
=
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
) holds
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) )
:|
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) )
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
finite
)
set
) ;
theorem
:: CLOSURE2:19
for
i
being ( ( ) ( )
set
)
for
f
,
f1
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) st
i
: ( ( ) ( )
set
)
in
dom
|.
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) ,
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) : ( ( ) ( )
set
) holds
|.
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) ,
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
)
.|
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) ,
(
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
finite
)
set
) ;
theorem
:: CLOSURE2:20
for
i
,
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
f
,
f1
being ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) ) st
i
: ( ( ) ( )
set
)
in
I
: ( ( ) ( )
set
) &
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) )
=
{
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
) ,
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
}
: ( ( ) ( non
empty
functional
finite
)
set
) holds
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
3
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) ) )
:|
: ( (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) ) (
Relation-like
b
2
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
2
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
2
: ( ( ) ( )
set
) )
.
i
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
=
{
(
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) ,
(
f1
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
Function
)
.
i
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
finite
)
set
) ;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
SF
be ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
:: original:
|:
redefine
func
|:
SF
:|
->
( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
I
: ( ( ) ( )
set
) ) ;
end;
theorem
:: CLOSURE2:21
for
I
being ( ( ) ( )
set
)
for
M
,
A
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
in
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
in'
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
b
1
: ( ( ) ( )
set
) ) ;
theorem
:: CLOSURE2:22
for
I
being ( ( ) ( )
set
)
for
M
,
A
,
B
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
=
{
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ,
B
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
}
: ( ( ) ( non
empty
functional
finite
)
set
) holds
union
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
b
1
: ( ( ) ( )
set
) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
=
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
\/
B
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) ;
theorem
:: CLOSURE2:23
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) st
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
=
{
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ,
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
}
: ( ( ) ( non
empty
functional
finite
)
set
) holds
meet
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
=
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
/\
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) ;
theorem
:: CLOSURE2:24
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
SF
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
for
Z
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st ( for
Z1
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) st
Z1
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
in
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
Z
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
c='
Z1
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
Z
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
c='
meet
|:
SF
: ( ( ) (
functional
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:25
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) holds
|:
(
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
V8
()
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
b
1
: ( ( ) ( )
set
) )
=
bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) )
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
ManySortedSubset
of
bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) ) ;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
IT
be ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
attr
IT
is
additive
means
:: CLOSURE2:def 4
for
A
,
B
being ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) st
A
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) &
B
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
A
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
\/
B
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
attr
IT
is
absolutely-additive
means
:: CLOSURE2:def 5
for
F
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) st
F
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
c=
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
union
|:
F
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
I
: ( ( ) ( )
set
) ) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
attr
IT
is
multiplicative
means
:: CLOSURE2:def 6
for
A
,
B
being ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) st
A
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) &
B
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
A
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
/\
B
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
attr
IT
is
absolutely-multiplicative
means
:: CLOSURE2:def 7
for
F
being ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) st
F
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
c=
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
meet
|:
F
: ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) )
:|
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
MSSubsetFamily
of
I
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSubset
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) )
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
attr
IT
is
properly-upper-bound
means
:: CLOSURE2:def 8
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
attr
IT
is
properly-lower-bound
means
:: CLOSURE2:def 9
[[0]]
I
: ( ( ) ( )
set
) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
empty-yielding
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
in
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
:: original:
Bool
redefine
func
Bool
M
->
( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
absolutely-additive
->
additive
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
absolutely-multiplicative
->
multiplicative
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
absolutely-multiplicative
->
properly-upper-bound
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
properly-upper-bound
->
non
empty
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
absolutely-additive
->
properly-lower-bound
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
properly-lower-bound
->
non
empty
for ( ( ) ( )
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
begin
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
mode
SetOp of
M
is
( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
Function
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ,
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
f
be ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
let
x
be ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) ;
:: original:
.
redefine
func
f
.
x
->
( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
IT
be ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
attr
IT
is
reflexive
means
:: CLOSURE2:def 10
for
x
being ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) holds
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
c='
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) ;
attr
IT
is
monotonic
means
:: CLOSURE2:def 11
for
x
,
y
being ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) st
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
c='
y
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) holds
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
c='
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
y
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) ;
attr
IT
is
idempotent
means
:: CLOSURE2:def 12
for
x
being ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) holds
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
=
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) ;
attr
IT
is
topological
means
:: CLOSURE2:def 13
for
x
,
y
being ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) holds
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
\/
y
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
)
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( ( ) ( )
set
)
=
(
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
\/
(
IT
: ( ( non
empty
) ( non
empty
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
y
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ) : ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ;
end;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
topological
for ( ( ) ( )
Element
of
bool
[:
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ,
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: CLOSURE2:26
for
I
being ( ( ) ( )
set
)
for
A
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) holds
id
(
Bool
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ) ( non
empty
Relation-like
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-defined
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-valued
Function-like
one-to-one
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
Function-yielding
V25
() )
Element
of
bool
[:
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ,
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( (
Function-like
quasi_total
reflexive
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
)
SetOp
of
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:27
for
I
being ( ( ) ( )
set
)
for
A
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) holds
id
(
Bool
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ) ( non
empty
Relation-like
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-defined
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-valued
Function-like
one-to-one
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
Function-yielding
V25
() )
Element
of
bool
[:
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ,
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( (
Function-like
quasi_total
monotonic
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
monotonic
)
SetOp
of
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:28
for
I
being ( ( ) ( )
set
)
for
A
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) holds
id
(
Bool
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ) ( non
empty
Relation-like
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-defined
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-valued
Function-like
one-to-one
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
Function-yielding
V25
() )
Element
of
bool
[:
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ,
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( (
Function-like
quasi_total
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
idempotent
)
SetOp
of
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:29
for
I
being ( ( ) ( )
set
)
for
A
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) holds
id
(
Bool
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ) ( non
empty
Relation-like
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-defined
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
-valued
Function-like
one-to-one
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
Function-yielding
V25
() )
Element
of
bool
[:
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ,
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) )
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) is ( (
Function-like
quasi_total
topological
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
topological
)
SetOp
of
A
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:30
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
for
g
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
=
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) &
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
reflexive
holds
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
=
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ;
theorem
:: CLOSURE2:31
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
g
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
reflexive
& ( for
X
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
X
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
c=
X
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ) holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
idempotent
;
theorem
:: CLOSURE2:32
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
for
g
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
for
A
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) st
A
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
=
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
/\
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) &
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
monotonic
holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
A
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
c=
(
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
/\
(
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) ;
registration
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
cluster
Function-like
quasi_total
topological
->
Function-like
quasi_total
monotonic
for ( ( ) ( )
Element
of
bool
[:
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ,
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: CLOSURE2:33
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
E
,
T
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
for
g
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
for
A
being ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) st
A
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
=
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
\
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
) &
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
topological
holds
(
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
E
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
\
(
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
T
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
functional
with_common_domain
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
set
)
c=
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
.
A
: ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
Element
of
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) ) ;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
let
h
,
g
be ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ) ;
:: original:
*
redefine
func
g
*
h
->
( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
set
) ) ;
end;
theorem
:: CLOSURE2:34
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
g
,
h
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
reflexive
&
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
reflexive
holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
reflexive
;
theorem
:: CLOSURE2:35
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
g
,
h
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
monotonic
&
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
monotonic
holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
monotonic
;
theorem
:: CLOSURE2:36
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
g
,
h
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
idempotent
&
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
idempotent
&
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
=
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
idempotent
;
theorem
:: CLOSURE2:37
for
I
being ( ( ) ( )
set
)
for
M
being ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) )
for
g
,
h
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
M
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) st
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
topological
&
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
topological
holds
g
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) )
*
h
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) )
quasi_total
)
SetOp
of
b
2
: ( (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) ) (
Relation-like
b
1
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
b
1
: ( ( ) ( )
set
) ) )
ManySortedSet
of
b
1
: ( ( ) ( )
set
) ) ) is
topological
;
begin
definition
let
S
be ( ( ) ( )
1-sorted
) ;
attr
c
2
is
strict
;
struct
ClosureStr
over
S
->
( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
aggr
ClosureStr
(#
Sorts
,
Family
#)
->
( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
sel
Family
c
2
->
( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
c
2
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
end;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
let
IT
be ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
attr
IT
is
additive
means
:: CLOSURE2:def 14
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
additive
;
attr
IT
is
absolutely-additive
means
:: CLOSURE2:def 15
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
absolutely-additive
;
attr
IT
is
multiplicative
means
:: CLOSURE2:def 16
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
multiplicative
;
attr
IT
is
absolutely-multiplicative
means
:: CLOSURE2:def 17
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
absolutely-multiplicative
;
attr
IT
is
properly-upper-bound
means
:: CLOSURE2:def 18
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
properly-upper-bound
;
attr
IT
is
properly-lower-bound
means
:: CLOSURE2:def 19
the
Family
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
IT
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
properly-lower-bound
;
end;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
func
Full
MS
->
( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
equals
:: CLOSURE2:def 20
ClosureStr
(# the
Sorts
of
MS
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
(
Bool
the
Sorts
of
MS
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
MS
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
Full
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( (
non-empty
) (
non-empty
)
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
Full
MS
: ( (
non-empty
) (
non-empty
)
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
strict
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
non-empty
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
cluster
non-empty
strict
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
for ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
additive
) (
additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
additive
) (
additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
additive
) (
additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
additive
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
absolutely-additive
) (
absolutely-additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
absolutely-additive
) (
absolutely-additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
absolutely-additive
) (
absolutely-additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
absolutely-additive
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
multiplicative
) (
multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
multiplicative
) (
multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
multiplicative
) (
multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
multiplicative
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
absolutely-multiplicative
) (
absolutely-multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
absolutely-multiplicative
) (
absolutely-multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
absolutely-multiplicative
) (
absolutely-multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
absolutely-multiplicative
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
properly-upper-bound
) (
properly-upper-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
properly-upper-bound
) (
properly-upper-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
properly-upper-bound
) (
properly-upper-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
properly-upper-bound
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
CS
be ( (
properly-lower-bound
) (
properly-lower-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
cluster
the
Family
of
CS
: ( (
properly-lower-bound
) (
properly-lower-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
CS
: ( (
properly-lower-bound
) (
properly-lower-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
->
properly-lower-bound
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
M
be ( (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ;
let
F
be ( ( ) (
functional
)
SubsetFamily
of
M
: ( (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ;
cluster
ClosureStr
(#
M
: ( (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( ( ) (
functional
)
Element
of
bool
(
Bool
M
: ( (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
V8
() the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
non-empty
strict
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
additive
) (
functional
additive
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
additive
) (
functional
additive
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
additive
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
absolutely-additive
) ( non
empty
functional
additive
absolutely-additive
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
absolutely-additive
) ( non
empty
functional
additive
absolutely-additive
properly-lower-bound
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
additive
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
absolutely-additive
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
multiplicative
) (
functional
multiplicative
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
multiplicative
) (
functional
multiplicative
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
multiplicative
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
absolutely-multiplicative
) ( non
empty
functional
multiplicative
absolutely-multiplicative
properly-upper-bound
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
absolutely-multiplicative
) ( non
empty
functional
multiplicative
absolutely-multiplicative
properly-upper-bound
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
multiplicative
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
absolutely-multiplicative
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
properly-upper-bound
) ( non
empty
functional
properly-upper-bound
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
properly-upper-bound
) ( non
empty
functional
properly-upper-bound
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
properly-upper-bound
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
MS
be ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) ;
let
F
be ( (
properly-lower-bound
) ( non
empty
functional
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
cluster
ClosureStr
(# the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ,
F
: ( (
properly-lower-bound
) ( non
empty
functional
properly-lower-bound
)
Element
of
bool
(
Bool
the
Sorts
of
MS
: ( ( ) ( )
many-sorted
over
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) ) #) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
properly-lower-bound
;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
cluster
absolutely-additive
->
additive
for ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
cluster
absolutely-multiplicative
->
multiplicative
for ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
cluster
absolutely-multiplicative
->
properly-upper-bound
for ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
cluster
absolutely-additive
->
properly-lower-bound
for ( ( ) ( )
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
mode
ClosureSystem of
S
is
( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) ) ;
end;
definition
let
I
be ( ( ) ( )
set
) ;
let
M
be ( (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) ) (
Relation-like
I
: ( ( ) ( )
set
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
set
) ) )
ManySortedSet
of
I
: ( ( ) ( )
set
) ) ;
mode
ClosureOperator of
M
is
( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
M
: ( (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
M
: ( (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
SetOp
of
M
: ( (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
I
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
I
: ( ( ) ( )
1-sorted
) ) )
set
) ) ;
end;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
let
A
be ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ;
let
g
be ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ;
func
ClOp->ClSys
g
->
( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
means
:: CLOSURE2:def 21
( the
Sorts
of
it
: ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ,
S
: ( ( ) ( )
1-sorted
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
=
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) & the
Family
of
it
: ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ,
S
: ( ( ) ( )
1-sorted
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
it
: ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ,
S
: ( ( ) ( )
1-sorted
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
=
{
x
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ) where
x
is ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) ) :
g
: ( (
properly-lower-bound
) ( non
empty
functional
properly-lower-bound
)
Element
of
bool
(
Bool
the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( ( ) ( non
empty
functional
with_common_domain
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) )
=
x
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
}
);
end;
registration
let
S
be ( ( ) ( )
1-sorted
) ;
let
A
be ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ;
let
g
be ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ;
cluster
ClOp->ClSys
g
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
Element
of
bool
[:
(
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ,
(
Bool
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
:]
: ( ( ) ( non
empty
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( (
strict
) (
strict
)
ClosureStr
over
S
: ( ( ) ( )
1-sorted
) )
->
strict
absolutely-multiplicative
;
end;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
let
A
be ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) ;
let
C
be ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSubset
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
func
Cl
C
->
( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
means
:: CLOSURE2:def 22
ex
F
being ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) st
(
it
: ( (
Function-like
quasi_total
) (
Relation-like
Function-like
quasi_total
)
Element
of
bool
[:
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ,
S
: ( ( ) ( )
1-sorted
)
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
=
meet
|:
F
: ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
:|
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
MSSubsetFamily
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSubset
of the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) &
F
: ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
=
{
X
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) where
X
is ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : (
C
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
Element
of
bool
[:
(
Bool
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) ,
(
Bool
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) )
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
c='
X
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) &
X
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
A
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
in
the
Family
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( ( ) (
functional
)
SubsetFamily
of the
Sorts
of
A
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
}
);
end;
theorem
:: CLOSURE2:38
for
S
being ( ( ) ( )
1-sorted
)
for
D
being ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) )
for
a
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
for
f
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) st
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
in
the
Family
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( ( ) ( non
empty
functional
multiplicative
absolutely-multiplicative
properly-upper-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) & ( for
x
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
.
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
Cl
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) ) holds
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
.
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) ;
theorem
:: CLOSURE2:39
for
S
being ( ( ) ( )
1-sorted
)
for
D
being ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) )
for
a
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
for
f
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) st
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
.
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) & ( for
x
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
.
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
Cl
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) ) holds
a
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
in
the
Family
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( ( ) ( non
empty
functional
multiplicative
absolutely-multiplicative
properly-upper-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ;
theorem
:: CLOSURE2:40
for
S
being ( ( ) ( )
1-sorted
)
for
D
being ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) )
for
f
being ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) st ( for
x
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
.
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
Cl
x
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) ) holds
(
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
reflexive
&
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
monotonic
&
f
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
)
SetOp
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) is
idempotent
) ;
definition
let
S
be ( ( ) ( )
1-sorted
) ;
let
D
be ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) ;
func
ClSys->ClOp
D
->
( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
means
:: CLOSURE2:def 23
for
x
being ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) holds
it
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
Element
of
bool
[:
(
Bool
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) ) ,
(
Bool
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
)
)
: ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) (
functional
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) )
:]
: ( ( ) (
Relation-like
)
set
) : ( ( ) ( non
empty
)
set
) )
.
x
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
=
Cl
x
: ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) : ( ( ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
Element
of
Bool
the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
D
: ( (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) ) (
Relation-like
S
: ( ( ) ( )
1-sorted
)
-defined
Function-like
V17
(
S
: ( ( ) ( )
1-sorted
) ) )
set
) : ( (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) ) ;
end;
theorem
:: CLOSURE2:41
for
S
being ( ( ) ( )
1-sorted
)
for
A
being ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
S
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) )
for
f
being ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
A
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) holds
ClSys->ClOp
(
ClOp->ClSys
f
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
(
ClOp->ClSys
b
3
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
(
ClOp->ClSys
b
3
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of the
Sorts
of
(
ClOp->ClSys
b
3
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) )
=
f
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of
b
2
: ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
ManySortedSet
of the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) ;
theorem
:: CLOSURE2:42
for
S
being ( ( ) ( )
1-sorted
)
for
D
being ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
S
: ( ( ) ( )
1-sorted
) ) holds
ClOp->ClSys
(
ClSys->ClOp
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) )
)
: ( (
Function-like
quasi_total
reflexive
monotonic
idempotent
) ( non
empty
Relation-like
Function-like
V17
(
Bool
the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) : ( (
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
) ( non
empty
functional
with_common_domain
additive
absolutely-additive
multiplicative
absolutely-multiplicative
properly-upper-bound
properly-lower-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) )
quasi_total
reflexive
monotonic
idempotent
)
ClosureOperator
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) : ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) )
=
ClosureStr
(# the
Sorts
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) , the
Family
of
D
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( ( ) ( non
empty
functional
multiplicative
absolutely-multiplicative
properly-upper-bound
)
SubsetFamily
of the
Sorts
of
b
2
: ( (
absolutely-multiplicative
) (
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureSystem
of
b
1
: ( ( ) ( )
1-sorted
) ) : ( (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) ) (
Relation-like
the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
)
-defined
Function-like
V17
( the
carrier
of
b
1
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) ) )
set
) ) #) : ( (
strict
) (
strict
multiplicative
absolutely-multiplicative
properly-upper-bound
)
ClosureStr
over
b
1
: ( ( ) ( )
1-sorted
) ) ;