:: PENCIL_3 semantic presentation
begin
theorem
:: PENCIL_3:1
for
S
being ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
)
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
S
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) )
for
p
,
q
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
are_collinear
holds
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) )
.
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) )
.
q
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
non
void
) ( non
empty
non
void
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
are_collinear
;
theorem
:: PENCIL_3:2
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
non-Trivial-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) holds not
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
1-sorted
) is
trivial
;
theorem
:: PENCIL_3:3
for
S
being ( ( non
void
identifying_close_blocks
) ( non
void
identifying_close_blocks
)
TopStruct
) st
S
: ( ( non
void
identifying_close_blocks
) ( non
void
identifying_close_blocks
)
TopStruct
) is
strongly_connected
holds
S
: ( ( non
void
identifying_close_blocks
) ( non
void
identifying_close_blocks
)
TopStruct
) is
without_isolated_points
;
theorem
:: PENCIL_3:4
for
S
being ( ( non
empty
non
void
identifying_close_blocks
) ( non
empty
non
void
identifying_close_blocks
)
TopStruct
) st
S
: ( ( non
empty
non
void
identifying_close_blocks
) ( non
empty
non
void
identifying_close_blocks
)
TopStruct
) is
strongly_connected
holds
S
: ( ( non
empty
non
void
identifying_close_blocks
) ( non
empty
non
void
identifying_close_blocks
)
TopStruct
) is
connected
;
theorem
:: PENCIL_3:5
for
S
being ( ( non
void
non
degenerated
) ( non
void
non
degenerated
)
TopStruct
)
for
L
being ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
S
: ( ( non
void
non
degenerated
) ( non
void
non
degenerated
)
TopStruct
) : ( ( ) ( )
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) holds
not for
x
being ( ( ) ( )
Point
of ( ( ) ( )
set
) ) holds
x
: ( ( ) ( )
Point
of ( ( ) ( )
set
) )
in
L
: ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
b
1
: ( ( non
void
non
degenerated
) ( non
void
non
degenerated
)
TopStruct
) : ( ( ) ( )
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: PENCIL_3:6
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
non-Empty
TopStruct-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
p
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) is ( ( ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
Element
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
non-Empty
TopStruct-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) ;
theorem
:: PENCIL_3:7
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
x
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) holds
(
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
)
.
x
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
)
=
[#]
(
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
1-sorted
) : ( ( ) ( non
proper
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
b
3
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
1-sorted
) : ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: PENCIL_3:8
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) ex
L
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
(
indx
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) &
product
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ) ;
theorem
:: PENCIL_3:9
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
p
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ex
L
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
(
indx
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) &
product
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) &
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
in
product
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) ) ;
theorem
:: PENCIL_3:10
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
b
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
product
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) holds
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
.
(
indx
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
)
=
[#]
(
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
(
indx
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
TopStruct
) : ( ( ) ( non
proper
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
(
indx
b
3
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
TopStruct
) : ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: PENCIL_3:11
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
L1
,
L2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
product
L1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) &
product
L2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) &
indx
L1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
indx
L2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) &
product
L1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
)
meets
product
L2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) holds
L1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
=
L2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
TopStruct-yielding
non-Trivial-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-Trivial-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) ;
theorem
:: PENCIL_3:12
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
L
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
for
B
being ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
(
indx
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) holds
product
(
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
+*
(
(
indx
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
B
: ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
(
indx
b
3
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) : ( ( ) ( )
set
) is ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: PENCIL_3:13
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
p
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
for
L
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
<>
indx
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) holds
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
+*
(
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
{
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
b
3
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) is ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) ;
theorem
:: PENCIL_3:14
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
S
being ( ( ) ( )
Subset
of )
for
L
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) holds
product
(
L
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
+*
(
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
S
: ( ( ) ( )
Subset
of ) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) : ( ( ) ( )
set
) is ( ( ) ( )
Subset
of ) ;
theorem
:: PENCIL_3:15
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
P
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) holds
{
P
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
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
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
trivial-yielding
Segre-like
)
set
)
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
) is 1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
;
theorem
:: PENCIL_3:16
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
B
being ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) )
for
P
being ( ( ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
Element
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) holds
product
(
{
P
: ( ( ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
Element
of
Carrier
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
}
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
trivial-yielding
Segre-like
)
set
)
+*
(
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
B
: ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
b
3
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
b
2
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) : ( ( ) ( )
set
) is ( ( ) ( )
Block
of ( ( ) ( non
empty
)
Element
of
bool
(
bool
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: PENCIL_3:17
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
p
,
q
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
<>
q
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
are_collinear
iff for
p1
,
q1
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
p1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) &
q1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
q
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
ex
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
( ( for
a
,
b
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
a
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
p1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
) &
b
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
=
q1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
) holds
(
a
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
<>
b
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) &
a
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
b
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
are_collinear
) ) & ( for
j
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
j
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
<>
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) holds
p1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
j
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
)
=
q1
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
j
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: PENCIL_3:18
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
b
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
for
x
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ex
p
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
(
p
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
in
product
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
{
(
p
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
+*
(
(
indx
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
x
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
)
}
: ( ( ) (
functional
non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
set
)
=
product
(
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
+*
(
(
indx
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ,
{
x
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
(
indx
b
3
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) : ( ( ) ( )
set
) ) ;
definition
let
I
be ( ( non
empty
finite
) ( non
empty
finite
)
set
) ;
let
b1
,
b2
be ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
func
diff
(
b1
,
b2
)
->
( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
equals
:: PENCIL_3:def 1
card
{
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) where
i
is ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) ) :
b1
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
)
<>
b2
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
)
}
: ( (
cardinal
) (
V21
()
V22
()
V23
()
cardinal
)
set
) ;
end;
theorem
:: PENCIL_3:19
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
b1
,
b2
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st
b1
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
)
<>
b2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
) holds
diff
(
b1
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ,
b2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
=
(
diff
(
b1
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ,
(
b2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
+*
(
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ,
(
b1
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
set
) )
)
: ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
+
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
begin
definition
let
I
be ( ( non
empty
) ( non
empty
)
set
) ;
let
A
be ( (
Relation-like
I
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) ;
let
B1
,
B2
be ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
I
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) ) ;
pred
B1
'||'
B2
means
:: PENCIL_3:def 2
for
x
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
x
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
in
B1
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) ) holds
ex
y
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
(
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
in
B2
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
) &
x
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ,
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
are_collinear
);
end;
theorem
:: PENCIL_3:20
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
B1
,
B2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
'||'
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
for
C1
,
C2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) st
C1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
=
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.:
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) &
C2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
=
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.:
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
C1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
'||'
C2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) ;
theorem
:: PENCIL_3:21
for
I
being ( ( non
empty
) ( non
empty
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
I
: ( ( non
empty
) ( non
empty
)
set
) )
for
B1
,
B2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
misses
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) holds
(
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
'||'
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) iff for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) holds
(
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
=
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) & ex
r
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
(
r
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
<>
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) & ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
<>
r
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) holds
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
.
i
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
set
)
=
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
.
i
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
set
) ) & ( for
c1
,
c2
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
.
r
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
)
=
{
c1
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
b
7
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) &
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) non
trivial-yielding
Segre-like
)
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) )
set
) )
.
r
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
)
=
{
c2
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
) ( non
empty
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
)
ManySortedSet
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
.
b
7
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
c1
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ,
c2
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
are_collinear
) ) ) ) ;
theorem
:: PENCIL_3:22
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
connected
) holds
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
for
p
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
=
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
+*
(
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ,
{
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
)
Element
of
bool
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
3
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Relation-like
Function-like
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) & not
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
in
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( non
empty
)
set
) holds
ex
D
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) st
(
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
)
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
len
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
)
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) & ( for
i
being ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) st
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
in
dom
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
finite
)
set
) ) holds
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) : ( ( ) ( )
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ) & ( for
i
being ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) st 1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
<=
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) &
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
<
len
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) holds
for
Di
,
Di1
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) st
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) : ( ( ) ( )
set
) &
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
+
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
) holds
(
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
misses
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
'||'
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ) ) ) ;
theorem
:: PENCIL_3:23
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
connected
) holds
for
B1
,
B2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
misses
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) holds
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) holds
(
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) iff ex
D
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) st
(
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
)
=
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
len
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
)
=
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) & ( for
i
being ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) st
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
in
dom
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
finite
)
set
) ) holds
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) : ( ( ) ( )
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ) & ( for
i
being ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) st 1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
<=
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) &
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
<
len
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) holds
for
Di
,
Di1
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) st
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
) : ( ( ) ( )
set
) &
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
D
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) )
-defined
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
finite
FinSequence-like
FinSubsequence-like
V259
() )
FinSequence
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
Nat
)
+
1 : ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
V21
()
V22
()
V23
() non
finite
cardinal
limit_cardinal
)
Element
of
bool
REAL
: ( ( ) ( )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
) holds
(
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
misses
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
Di
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
'||'
Di1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ) ) ) ) ;
theorem
:: PENCIL_3:24
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
for
B1
,
B2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
for
b1
,
b2
,
b3
,
b4
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.:
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
product
b3
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.:
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
product
b4
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
indx
b3
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
indx
b4
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
theorem
:: PENCIL_3:25
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ex
s
being ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st
for
i
,
j
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
(
s
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
j
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) iff for
B1
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.:
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
j
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ;
definition
let
I
be ( ( non
empty
finite
) ( non
empty
finite
)
set
) ;
let
A
be ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
assume
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
;
let
f
be ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ;
func
permutation_of_indices
f
->
( (
Function-like
quasi_total
bijective
) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
I
: ( ( ) ( )
TopStruct
)
-valued
Function-like
one-to-one
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( ) ( )
TopStruct
) )
means
:: PENCIL_3:def 3
for
i
,
j
being ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) ) holds
(
it
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
=
j
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) iff for
B1
being ( ( ) ( )
Segre-Coset
of
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) )
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) ) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) )
set
) ) st
B1
: ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
=
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
.:
B1
: ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
Segre_Product
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( non
empty
) ( non
empty
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) &
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
=
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
=
j
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) );
end;
begin
definition
let
I
be ( ( non
empty
finite
) ( non
empty
finite
)
set
) ;
let
A
be ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
assume
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
;
let
f
be ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ;
let
b1
be ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ;
assume
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) ;
func
canonical_embedding
(
f
,
b1
)
->
( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
indx
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
(
permutation_of_indices
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
I
: ( ( ) ( )
TopStruct
)
-valued
Function-like
one-to-one
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( ) ( )
TopStruct
) )
.
(
indx
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
indx
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
means
:: PENCIL_3:def 4
(
it
: ( ( ) ( )
Element
of
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) is
isomorphic
& ( for
a
being ( (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) ) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
V259
() )
ManySortedSet
of
I
: ( ( ) ( )
TopStruct
) ) st
a
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) is ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) &
a
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
in
product
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
) : ( ( ) ( )
set
) holds
for
b
being ( (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) ) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
V259
() )
ManySortedSet
of
I
: ( ( ) ( )
TopStruct
) ) st
b
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
.
a
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
) holds
b
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
(
permutation_of_indices
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
I
: ( ( ) ( )
TopStruct
)
-valued
Function-like
one-to-one
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( ) ( )
TopStruct
) )
.
(
indx
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) ) : ( ( ) ( )
set
)
=
it
: ( ( ) ( )
Element
of
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) )
.
(
a
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b1
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) );
end;
theorem
:: PENCIL_3:26
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
for
B1
,
B2
being ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) st
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
misses
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) )
'||'
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) holds
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
)
=
B1
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
)
=
B2
: ( ( ) ( )
Segre-Coset
of
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) holds
canonical_embedding
(
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ,
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
6
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
(
permutation_of_indices
b
3
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
6
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
6
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
=
canonical_embedding
(
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ,
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
7
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
(
permutation_of_indices
b
3
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
7
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
7
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) ;
theorem
:: PENCIL_3:27
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
for
b1
,
b2
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) st
product
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
product
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) &
indx
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
indx
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
canonical_embedding
(
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ,
b1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
4
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
(
permutation_of_indices
b
3
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
4
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
4
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
=
canonical_embedding
(
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ,
b2
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
5
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
(
permutation_of_indices
b
3
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
5
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
indx
b
5
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) ;
definition
let
I
be ( ( non
empty
finite
) ( non
empty
finite
)
set
) ;
let
A
be ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
assume
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
;
let
f
be ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ;
let
i
be ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ;
func
canonical_embedding
(
f
,
i
)
->
( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
(
permutation_of_indices
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
I
: ( ( ) ( )
TopStruct
)
-valued
Function-like
one-to-one
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( ) ( )
TopStruct
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
)
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
means
:: PENCIL_3:def 5
for
b
being ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) ) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
Function-like
V14
(
I
: ( ( ) ( )
TopStruct
) ) )
set
) ) st
product
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( non
empty
non
trivial
)
set
) is ( ( ) ( )
Segre-Coset
of
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) &
indx
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) : ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
=
i
: ( (
V27
() ) (
V21
()
V22
()
V23
()
V27
()
finite
cardinal
V66
()
V230
()
ext-real
non
negative
)
set
) holds
it
: ( ( ) ( )
Element
of
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) )
=
canonical_embedding
(
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) ) ,
b
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
indx
b
1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
(
permutation_of_indices
f
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
I
: ( ( ) ( )
TopStruct
)
-defined
I
: ( ( ) ( )
TopStruct
)
-valued
Function-like
one-to-one
non
empty
V14
(
I
: ( ( ) ( )
TopStruct
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( ) ( )
TopStruct
) )
.
(
indx
b
1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
A
: ( ( ) ( )
Element
of
bool
(
bool
I
: ( ( ) ( )
TopStruct
)
)
: ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
.
(
indx
b
1
: ( ( non
trivial-yielding
Segre-like
) (
Relation-like
non-empty
non
empty-yielding
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) non
trivial-yielding
Segre-like
V259
() )
ManySortedSubset
of
Carrier
A
: ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
set
) )
)
: ( ( ) ( )
Element
of
I
: ( ( ) ( )
TopStruct
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: PENCIL_3:28
for
I
being ( ( non
empty
finite
) ( non
empty
finite
)
set
)
for
A
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st ( for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) : ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) is
strongly_connected
) holds
for
f
being ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
A
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) ) ex
s
being ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ex
B
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st
for
i
being ( ( ) ( )
Element
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) holds
(
B
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
) is ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
b
4
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) & ( for
m
being ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
b
4
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) st
m
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
=
B
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
) holds
m
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) is
isomorphic
) & ( for
p
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
for
a
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st
a
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
for
b
being ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
I
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) st
b
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
=
f
: ( (
Function-like
quasi_total
isomorphic
) (
Relation-like
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
isomorphic
)
Collineation
of
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) )
.
p
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
(
Segre_Product
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
for
m
being ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
b
4
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) ) st
m
: ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
b
4
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
=
B
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
Function-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
) holds
b
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
s
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) : ( ( ) ( )
set
)
=
m
: ( (
Function-like
quasi_total
) (
Relation-like
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
(
b
4
: ( (
Function-like
quasi_total
bijective
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-valued
Function-like
one-to-one
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
quasi_total
onto
bijective
V259
() )
Permutation
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
non
empty
V14
( the
carrier
of
(
b
2
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
PLS-yielding
) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
1-sorted-yielding
non-Empty
TopStruct-yielding
non-void-yielding
non-Trivial-yielding
PLS-yielding
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
b
6
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
) ( non
empty
non
void
non
degenerated
with_non_trivial_blocks
identifying_close_blocks
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
quasi_total
)
Function
of ( ( ) ( non
empty
)
set
) , ( ( ) ( non
empty
)
set
) )
.
(
a
: ( (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) ) ) (
Relation-like
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
)
-defined
Function-like
non
empty
V14
(
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
V259
() )
ManySortedSet
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
.
i
: ( ( ) ( )
Element
of
b
1
: ( ( non
empty
finite
) ( non
empty
finite
)
set
) )
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ;