:: SPRECT_2 semantic presentation
begin
theorem
:: SPRECT_2:1
for
i
,
j
,
k
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
+
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
-'
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:2
for
i
,
j
,
k
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
>
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
-'
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
+
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:3
for
i
,
j
,
k
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
/.
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
=
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
/.
(
(
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
+
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
-'
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:4
for
i
,
j
,
k
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
>
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
/.
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
=
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) )
/.
(
(
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
-'
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
+
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
4
: ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:5
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
len
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
>=
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:6
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
len
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
=
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) holds
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:7
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
not
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) is
empty
;
theorem
:: SPRECT_2:8
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
=
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
/.
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:9
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
D
being ( ( non
empty
) ( non
empty
)
set
)
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
D
: ( ( non
empty
) ( non
empty
)
set
) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
/.
(
len
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
=
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) )
/.
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
b
3
: ( ( non
empty
) ( non
empty
)
set
) ) ;
begin
theorem
:: SPRECT_2:10
for
X
being ( (
compact
) (
compact
)
Subset
of )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
X
: ( (
compact
) (
compact
)
Subset
of ) &
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
N-bound
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) holds
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
N-most
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:11
for
X
being ( (
compact
) (
compact
)
Subset
of )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
X
: ( (
compact
) (
compact
)
Subset
of ) &
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
S-bound
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) holds
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
S-most
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:12
for
X
being ( (
compact
) (
compact
)
Subset
of )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
X
: ( (
compact
) (
compact
)
Subset
of ) &
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
W-bound
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) holds
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
W-most
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:13
for
X
being ( (
compact
) (
compact
)
Subset
of )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
X
: ( (
compact
) (
compact
)
Subset
of ) &
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
E-bound
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) holds
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
E-most
X
: ( (
compact
) (
compact
)
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
begin
theorem
:: SPRECT_2:14
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) holds
L~
(
mid
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
=
union
{
(
LSeg
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ,
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) where
k
is ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : (
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
k
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
}
: ( ( ) ( )
set
) ;
theorem
:: SPRECT_2:15
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
dom
(
X_axis
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
V127
()
V128
()
V129
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
=
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:16
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
dom
(
Y_axis
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
(
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
V127
()
V128
()
V129
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
=
dom
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:17
for
a
,
b
,
c
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
LSeg
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) &
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) & not
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) & not
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) ;
theorem
:: SPRECT_2:18
for
a
,
b
,
c
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
LSeg
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) &
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) & not
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) & not
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) ;
theorem
:: SPRECT_2:19
for
a
,
b
,
c
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
LSeg
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) &
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
>=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
>=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) & not
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) & not
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) ;
theorem
:: SPRECT_2:20
for
a
,
b
,
c
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
in
LSeg
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) &
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
>=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
>=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) & not
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) & not
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
=
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
a
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
c
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
b
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) ;
begin
definition
let
f
,
g
be ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
pred
g
is_in_the_area_of
f
means
:: SPRECT_2:def 1
for
n
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
W-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
E-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
S-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<=
N-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) );
end;
theorem
:: SPRECT_2:21
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:22
for
f
,
g
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
mid
(
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:23
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
mid
(
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:24
for
f
,
g
,
h
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) &
h
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
g
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
^
h
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:25
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
<*
(
NE-corner
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:26
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
<*
(
NW-corner
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:27
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
<*
(
SE-corner
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:28
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
<*
(
SW-corner
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
is_in_the_area_of
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
begin
definition
let
f
,
g
be ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
pred
g
is_a_h.c._for
f
means
:: SPRECT_2:def 2
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
is_in_the_area_of
f
: ( ( ) ( )
RLTopStruct
) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
W-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
(
len
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
E-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) );
pred
g
is_a_v.c._for
f
means
:: SPRECT_2:def 3
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
is_in_the_area_of
f
: ( ( ) ( )
RLTopStruct
) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
S-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) &
(
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
/.
(
len
g
: ( ( ) ( )
Element
of
f
: ( ( ) ( )
RLTopStruct
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
N-bound
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) );
end;
theorem
:: SPRECT_2:29
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
for
g
,
h
being ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st 2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
g
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & 2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
h
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
g
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_a_h.c._for
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) &
h
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
is_a_v.c._for
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
L~
g
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
meets
L~
h
: ( (
one-to-one
special
) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
begin
definition
let
f
be ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
attr
f
is
clockwise_oriented
means
:: SPRECT_2:def 4
(
Rotate
(
f
: ( ( ) ( )
RLTopStruct
) ,
(
N-min
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
/.
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
N-most
(
L~
f
: ( ( ) ( )
RLTopStruct
)
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
end;
theorem
:: SPRECT_2:30
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) is
clockwise_oriented
iff
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
N-most
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
registration
cluster
R^2-unit_square
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
->
compact
;
end;
theorem
:: SPRECT_2:31
N-bound
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:32
W-bound
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
0
: ( ( ) (
empty
trivial
natural
V11
()
real
Function-like
functional
FinSequence-membered
V37
()
ext-real
non
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:33
E-bound
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:34
S-bound
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
0
: ( ( ) (
empty
trivial
natural
V11
()
real
Function-like
functional
FinSequence-membered
V37
()
ext-real
non
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:35
N-most
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
=
LSeg
(
|[
0
: ( ( ) (
empty
trivial
natural
V11
()
real
Function-like
functional
FinSequence-membered
V37
()
ext-real
non
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
]|
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
Function-like
V26
()
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
V127
()
V128
()
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
|[
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
]|
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
Function-like
V26
()
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
V127
()
V128
()
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:36
N-min
R^2-unit_square
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
|[
0
: ( ( ) (
empty
trivial
natural
V11
()
real
Function-like
functional
FinSequence-membered
V37
()
ext-real
non
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
]|
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
Function-like
V26
()
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
V127
()
V128
()
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
registration
let
X
be ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) ;
cluster
SpStSeq
X
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
->
clockwise_oriented
;
end;
registration
cluster
non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
for ( ( ) ( )
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: SPRECT_2:37
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
>
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) or ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) ) holds
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:38
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) or ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) ) holds
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:39
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
N-min
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:40
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
N-max
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:41
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
S-min
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:42
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
S-max
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:43
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
W-min
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:44
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
W-max
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:45
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
E-min
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:46
for
f
being ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
E-max
(
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
rng
f
: ( ( non
trivial
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:47
for
i
,
j
,
m
,
n
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) or
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) holds
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
misses
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:48
for
i
,
j
,
m
,
n
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) or
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) holds
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
misses
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:49
for
i
,
j
,
m
,
n
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) or
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) holds
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
misses
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:50
for
i
,
j
,
m
,
n
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) &
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<=
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) & ( 1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) or
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) holds
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
misses
L~
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
m
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:51
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
(
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<
(
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ;
theorem
:: SPRECT_2:52
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:53
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
(
E-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<
(
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ;
theorem
:: SPRECT_2:54
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
E-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:55
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
(
S-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<
(
S-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ;
theorem
:: SPRECT_2:56
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
S-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
S-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:57
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
(
W-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
<
(
W-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ;
theorem
:: SPRECT_2:58
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
W-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
W-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_2:59
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) holds
LSeg
(
(
NW-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
misses
LSeg
(
(
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ;
theorem
:: SPRECT_2:60
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) is
being_S-Seq
&
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
<>
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) & (
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) or
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) &
(
LSeg
(
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
/\
(
L~
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
=
{
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) holds
<*
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:61
for
f
being ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
for
p
being ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) is
being_S-Seq
&
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
<>
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
(
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) & (
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
(
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) or
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) )
=
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
(
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) ) ) &
(
LSeg
(
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
(
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V218
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
/\
(
L~
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
=
{
(
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
(
len
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) holds
f
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
^
<*
p
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Point
of ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
begin
theorem
:: SPRECT_2:62
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
<*
(
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:63
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
<*
(
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:64
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
S-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
S-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
SE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
<*
(
SE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:65
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
E-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
<*
(
NE-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:66
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
NW-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
<*
(
NW-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
theorem
:: SPRECT_2:67
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
for
i
,
j
being ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
in
dom
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) ( non
empty
non
trivial
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
bool
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) &
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
W-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
W-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
SW-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
<*
(
SW-corner
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
V33
(1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
special
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
(
mid
(
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ,
i
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ,
j
: ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) (
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V26
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) is ( (
being_S-Seq
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
one-to-one
V26
()
FinSequence-like
FinSubsequence-like
special
unfolded
s.n.c.
being_S-Seq
)
S-Sequence_in_R2
) ;
registration
let
f
be ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ;
cluster
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
->
being_simple_closed_curve
;
end;
begin
theorem
:: SPRECT_2:68
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:69
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
N-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
>
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:70
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
N-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
E-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
N-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
E-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:71
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
E-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
E-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:72
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
E-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
S-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
E-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
S-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:73
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
S-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
S-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:74
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
S-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
W-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
S-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
W-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:75
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
<>
W-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
W-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
(
W-max
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:76
for
z
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) st
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
W-min
(
L~
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
z
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
clockwise_oriented
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: SPRECT_2:77
for
f
being ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) st
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
/.
1 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
(
W-max
(
L~
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
)
)
: ( ( ) ( non
empty
compact
being_simple_closed_curve
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) )
)
: ( ( ) (
V33
(2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
V129
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
..
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
<
len
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V13
()
V16
(
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
V17
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
natural
V11
()
real
V37
()
ext-real
positive
non
negative
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V103
()
V149
()
V150
()
V151
()
V152
()
V153
()
V154
()
V155
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V26
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) : ( ( ) (
natural
V11
()
real
V37
()
ext-real
V90
()
V137
()
V138
()
V139
()
V140
()
V141
()
V142
() )
Element
of
NAT
: ( ( ) (
V137
()
V138
()
V139
()
V140
()
V141
()
V142
()
V143
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V26
()
V137
()
V138
()
V139
()
V143
() )
set
) : ( ( ) ( )
set
) ) ) ;